Other than what TLC has covered, religion is a sensitive issue in Westeros.
Majority of the people in Westeros adhere to the Faith and followers of other religions are in minority, specifically limited to the North and Iron Islands disregarding small exceptions like Blackwoods in Riverlands and Manderleys in the North. Ever since Aegon took the throne and converted to the Faith, it has been generally accepted that the King must be from the majority religion i.e. the Faith. It is implied due to the fact that Northmen and Ironborn kept their vows for most of the times. That can be attributed to the internal autonomy of the both regions, neither of them answer to anyone other than the King and are free to rule themselves as they wish as long as they keep faith with the Crown and do not break the King's laws. These responsibilities lie on shoulders of the Lords Paramounts of both regions.
But that does not mean that People in Westeros do not generally care about the religion.
When King Aenys I tried to marry his daughter to his son, he was denounced as "King abomination" by the High Septon and was soon fighting a rebellion. This Valyrian tradition was not welcome at all in the faith of the Seven. Lords and Commons alike took sides with thousands rallying to the Sept to protect the Faith. It would end only in the reign of King Jaehaerys I who disbanded the faith militant and vowed that the Crown shall always defend the Holy Mother Sept, a vow all Targaryens kept.
Some counselors urged the Old King to deal with the remnants of the
Faith Militant harshly—to stamp them out once and for all before their
zealotry could return the realm to chaos. Others cared more for
ensuring that the septons were answerable to the same justice as the
rest of the realm. But Jaehaerys instead dispatched Septon Barth to
Oldtown, to speak with the High Septon, and there they began to forge
a lasting agreement. In return for the last few Stars and Swords
putting down their weapons, and for agreeing to accept outside
justice, the High Septon received King Jaehaerys's sworn oath that the
Iron Throne would always protect and defend the Faith. In this way,
the great schism between crown and Faith was forever healed.
TWOIAF -Targaryen Kings : Jaehaerys I
So while the Faith is happy to be peaceful as long as a follower of the Faith, in this case the King, guarantees to always defend the Faith against religious threats, why should they believe vows of a non-believer to be genuine? If someone who kept other gods than the Seven took the throne, that would force the Sept to reconsider their arrangement with the Crown naturally. How long before the heathen outlaws their one true faith eh? Or persecutes them? Or forces people to convert to their superstition?
The religious minorities were forced to tolerate the Faith once Aegon converted and took the throne, given the Targaryen power.
And since Aegon had accepted the Seven as his gods and been anointed
by the High Septon in Oldtown, Lord Vickon allowed the septons to
return to the islands once again to preach the Faith.
This angered many pious ironborn and provoked the wroth of the priests of the Drowned God, as it always had before. "Let them
preach," Lord Vickon said, when told of the unrest. "We have need of
winds to fill our sails." He was Aegon's man, he reminded his son
Goren, and no man but a fool would dare rise against Aegon Targaryen
and his dragons.
TWOIAF -The Iron Islands : The Greyjoys of Pyke
They let them preach because as long as the Throne did not force the conversion, there was no reason to risk a confrontation with the Crown. But Goren got rid of the Faith soon.
These were words that Goren Greyjoy would remember. When Lord Vickon
died in 33 AC, Goren succeeded him as Lord of the Iron Islands,
putting down a clumsy conspiracy to restore the black line by crowning
Qhorin Volmark's son in his stead. He faced a more serious test four
years later, when Aegon the Conqueror died of a stroke on Dragonstone,
and his son Aenys was crowned king in his stead. Though amiable and
well-meaning, Aenys Targaryen was widely perceived as a weakling,
unfit to sit the Iron Throne. The new king was still on his royal
progress when rebellions began to break out all across the realm. One
such revolt convulsed the Iron Islands, led by a man claiming that he
was the priestking Lodos returned at last from visiting his father.
But Goren Greyjoy dealt with it decisively, going so far as to send
the priestking's pickled head to Aenys Targaryen. His Grace was so
pleased with the gift that he promised Lord Goren any boon that was
within his power to grant. As sage as he was savage, Greyjoy asked
the king to give him leave to expel the septons and septas from the
Iron Islands. King Aenys was forced to agree. A century would pass
before another sept was opened on the islands.
TWOIAF -The Iron Islands : The Greyjoys of Pyke
While the Septons did return, that shows that the minorities, while willing to accept the Crown's authority, were not okay with influence of the other religions on them. If minorities feel that way, Majority would feel that as well and more strongly if follower of some other faith was to rule them.
Lannisters used exactly that against Stannis when he took the faith of the Red Priests.
Tyrion made a show of glancing over the writing again. There had been
some niggling phrase . . . "Done in the Light of the Lord," he read.
"A queer choice of words, that."
Pycelle cleared his throat. "These words often appear in letters and
documents from the Free Cities. They mean no more than, let us say,
written in the sight of god. The god of the red priests. It is their
usage, I do believe."
"Varys told us some years past that Lady Selyse had taken up with a
red priest," Littlefinger reminded them.
Tyrion tapped the paper. "And now it would seem her lord husband has
done the same. We can use that against him. Urge the High Septon to
reveal how Stannis has turned against the gods as well as his rightful
king . . ."
ACOK - Tyrion III
Since the High Septon is considered to be voice of the gods on Earth by the majority, his words will carry great weight.
Cersei tried to use the same support for Tommen:
“Most have lost their homes. Suffering is everywhere... and grief, and
death. Before coming to King’s Landing, I tended to half a hundred
little villages too small to have a septon of their own. I walked from
each one to the next, performing marriages, absolving sinners of their
sins, naming newborn children. Those villages are no more, Your Grace.
Weeds and thorns grow where gardens once flourished, and bones litter
“War is a dreadful thing. These atrocities are the work of the
northmen, and of Lord Stannis and his demon-worshipers.”
“The realm is full of kings. For the Faith to exalt one above the
rest we must be certain. Three hundred years ago, when Aegon the
Dragon landed beneath this very hill, the High Septon locked himself
within the Starry Sept of Oldtown and prayed for seven days and seven
nights, taking no nourishment but bread and water. When he emerged he
announced that the Faith would not oppose Aegon and his sisters, for
the Crone had lifted up her lamp to show him what lay ahead. If
Oldtown took up arms against the Dragon, Oldtown would burn, and the
Hightower and the Citadel and the Starry Sept would be cast down and
destroyed. Lord Hightower was a godly man. When he heard the prophecy,
he kept his strength at home and opened the city gates to Aegon when
he came. And His High Holiness anointed the Conqueror with the seven
oils. I must do as he did, three hundred years ago. I must pray, and
“For seven days and seven nights?”
“For as long as need be.”
Cersei itched to slap his solemn, pious face. I could help you fast,
she thought. I could shut you up in some tower and see that no one
brings you food until the gods have spoken. “These false kings
espouse false gods,” she reminded him. “Only King Tommen defends
the Holy Faith.”
AFFC - Cersei VI
But Zealotry is a two-way street. Luckily, most of the Kings, even Aerys II, had the wisdom to tread carefully in the matters of faith and did not vex the religious minorities in the North and Iron Isles. Baelor the Blessed was however not that wise and in his reign, the relations were set to be soured between the Crown and the minorities due to his zealotry. Thanks to the alleged intervention of his uncle Viserys, the King could not begin his project to force conversion of all his subjects and the Targaryen dynasty did not face a revolt in the North and the West.
Malicious rumors that followed in the wake of Viserys's
ascension—begun, some say, by the pen of the Lady Maia of House
Stokeworth—suggested that Viserys poisoned the king in order to
finally gain the throne after a decade and more of waiting. Others
have suggested that Viserys poisoned Baelor for the good of the
realm, since the septon-king had come to believe that the Seven called
on him to convert all the unbelievers in his realm. This would have
led to a war with the North and the Iron Islands that would have
caused great turmoil.
TWOIAF -Targaryen Kings : Baelor I
So in conclusion, yes it definitely matters which religion does the King follow. The King is the defender of Holy Mother Sept and therefore cannot espouse any other religion if he is to be seen that way. Furthermore, given the demographics of the realm, it is imperative that the King must have the same religion as the majority of the People. That being said, it is also equally important for the King to refrain from antagonizing the powerful minorities in the West and North and act like a symbol of unity for all the Westerosi people, regardless to their faith.