Why don’t people fight over religion in the universe of A Song Of Ice And Fire, like they do in ours?

Question

How come, in a world that can be described as barbaric and cruel and pretty much similar to the medieval ages in our universe, people who pray for the old gods and the new gods are fine with that, whereas they do share our thirst for battle over other things such as belonging to two ancient houses that were involved in feuds over simple things?

Apart from the followers of the lord of light R'hllor (that if you ask me is a clear parallel to Islam as it shares its provenance from the east and the radical followers), all other religions live in peace, and we don't even feel the discrimination by religion between people.

I will try to reformulate the question to be more precise: are there any mentions of past feuds in Westeros over religions and faiths?

Answer 1

Neither of the two major religions in Westeros have a strong emphasis on proselytizing. The faith in the Old Gods is held by the people of the north and those beyond the wall. It is somewhat of given since for practical purposes, the Weirwood trees have been there for 6000 years. As the faith is tied to the trees, there is no motivation to expand the faith beyond the areas where the trees grow.

The New Gods are imported from Essos, but that was a few thousand years ago also. There does not seem to be any evidence in cannon of Septims being ordained as missionaries and trying to go north to convert the northerners to the New Gods.

Overall the attitude toward religion is far more alike that of the Pre-Christian Ancient Near East, and the Pre-Christian world in general. When you met an enemy that had a different faith, it was your gods against theirs. Your success on the battlefield was directly tied to the success of your gods. When you made peace, the faith's mingled; some believed some gods, some believed in others, and it was not really a big issue (Except for a few "jealous" gods).

During the Medieval European era, political motivations led the Catholic Church to wage a war for the trade routes on the eastern coast of the Mediterranean Sea. This land was valuable to the Babylonian Empire, and the Assyrian Empire, and to Alexander the Great, and the Ptolemies and the Seleucids. The church used the guise of liberating the "Holy Land" from heathens and heretics to gain access to the trade routes.

The politics of Westeros are, to some degree, separate from the religion of the people. The Western, Southern, and Eastern domains follow the Seven Faces of God, as does King's Landing. It would not be effective for a Lannister to try to rally support against a Tyrell under the banner of the Seven since both sides are faithful. It may be possible for the Southern domains to unite under the banner of the Seven and march north to cut down all of the Weirwoods and eradicate the faith in the Old Gods, but no one seems motivated to do that.

The phrase of prayer "the Old Gods and the New" is spoken by many people in various locations some more devout than others. It shows the underlying feeling of most people is that they will follow any gods so long as they give the person what he or she is praying for, and that in desperate times either or both are acceptable. It could almost be said that all of Westeros is one religion in transition between the Old Gods and the New, and that nearly everyone believes in both, just to varying degrees.

Answer 2

There are definitely mentions of feuds over religious issues aside from the activities of the followers of R'hllor. The earliest example is the war between the Children of the Forest and the First Men, which began after the First Men cut down the sacred weirwood trees. In another example, The World Of Ice and Fire describes an outright war between the Faith Militant and the early Targaryen kings over their customs of polygamy and incest.

Answer 3

The other answers already provide plenty of material, in mine I'll focus on one particular conflict: religion on the Iron Islands. This is sourced from The World of Ice & Fire: the Untold History of Westeros and the Game of Thrones by George R.R. Martin, Elio Garcia, and Linda Antonsson.

The Iron Islands worshiped the Drwoned God, until the Andals invaded. This invasion shook up the existing power structures and the Iron Islands ended up with a new king, Harras Hoare. This was the start of an internal struggle in which religion was a significant factor:

Archmaester Hake tells us that the kings of House Hoare were, “black of hair, black of eye, and black of heart.” Their foes claimed their blood was black as well, darkened by the “Andal taint,” for many of the early Hoare kings took maidens of that ilk to wife. True ironborn had salt water in their veins, the priests of the Drowned God proclaimed; the black-blooded Hoares were false kings, ungodly usurpers who must be cast down.

[...]

Were the kings of House Hoare truly as ungodly as these holy men proclaimed? Hake believes they were, but Archmaester Haereg takes a very different view, suggesting that the true crime of the “black-blooded” kings was neither impiety nor demon-worship, but tolerance. For it was under the Hoares that the Faith of the Andals came to the Iron Islands for the first time.

We then see a major uprising in which religion plays a significant role:

Prompted by their Andal queens, these kings granted the septas and septons their protection and gave them leave to move about the islands, preaching of the Seven. The first sept on the Iron Islands was built on Great Wyk during the reign of Wulfgar Widowmaker. When his great-grandson Horgan permitted the building of another on Old Wyk, where the kingsmoots had been held of old, the entire island rose up in bloody rebellion, goaded by the priests. The sept was burned, the septon pulled to pieces, the worshippers dragged into the sea to drown, that they might regain their faith. It was in answer to this, Haereg alleges, that Horgan Hoare began to slaughter priests.

One of the descendants of this king ran into problems when he tried to extinguish some of the traditional customs of the Ironborn:

His own sons were raised in the Faith, or King Harmund’s own peculiar version of it. Upon his death, the eldest of them ascended the throne. Harmund the Handsome (influenced, some say, by his Lannister mother, the Dowager Queen Lelia) announced that henceforth reavers would be hanged as pirates rather than celebrated, and formally outlawed the taking of salt wives, declaring the children of such unions to be bastards with no right of inheritance. He was considering a measure to end the practice of thralldom on the isles as well when a priest known as the Shrike began to preach against him.

Other priests took up the cry, and the lords of the isles took heed. Only the septons and their followers stood by King Harmund, and he was overthrown within a fortnight, almost bloodlessly. What followed was far from bloodless, however. The Shrike himself tore out the deposed king’s tongue, so he might never again speak “lies and blasphemies.” Harmund was blinded as well, and his nose was cut off, so “all men might see him for the monster he is.”

In his place, the lords and priests crowned his younger brother Hagon. The new king denounced the Faith, rescinded Harmund’s edicts, and expelled the septons and septas from his realm. Within a fortnight every sept in the Iron Islands was aflame.

King Hagon, soon to be known as Hagon the Heartless, even permitted the mutilation of his own mother, Queen Lelia, the Lannister “whore” who was blamed by the Shrike for turning her husband and sons away from the true god.

This infighting started the decline of the Iron Islands, and lasted until Aegon's campaign:

Aegon the Conqueror put an end to the fighting in 2 AC when he and Balerion descended upon Great Wyk, accompanied by a vast war fleet. [...] Gathering the remaining lords of the Iron Islands together, he announced that he would allow them to choose their own lord paramount. Unsurprisingly they chose one of their own: Vickon Greyjoy, Lord Reaper of Pyke, a famous captain descended of the Grey King.

[...]

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.”

It wasn't until Balon Greyjoy that the Faith of the Seven lost influence:

Yet even as a child, Lord Balon had burned to free the ironborn from the yoke of the Iron Throne and restore them to a place of pride and power. Once seated on the Seastone Chair, he swept away many of his lord father’s decrees, abolishing the taxes on salt wives and declaring that men taken captive in war could indeed be kept as thralls. Though he did not expel the septons, he increased the taxes on them tenfold.