TL; DR: No. Not only have Arnor and Rohan always been two distinct entities, they had never existed at the same time until Aragorn became king of Gondor and Arnor.
Flaws in the question:
As the new High King, Aragorn took over and combined the 'kingship' from the Steward of Gondor and King Théoden of Rohan.
No, Aragorn combined the kingship of Gondor with the long-extinct kingship of the long-defunct kingdom of Arnor. Rohan remained independent, and Éomer succeeded Théoden as the King of Rohan.
As we can see in Tolkien's map of Middle-earth, Arnor (or "The Northern Kingdom") is in the northwest - in the same neighborhood as the Shire - and Rohan (or "The Horse Country") is much rather south and east.
The better-known map tells the same story.
And in Karen Wynn Fonstad's The Atlas of Middle-Earth, we see that after Aragorn took the throne of Gondor and Arnor, he reunited the two kingdoms (minus the Shire and a few other areas); Rohan, however, remained independent (but allied to the Reunited Kingdom).
The original kingdom of Arnor, established in the year 3320 of the Second Age, had been divided into the three separate (and often mutually hostile) petty kingdoms of Arthedain, Cardolan, and Rhudaur around 861 TA, which made the "Kingdom of Arnor" more a fond memory than a reality. When the last of these three kingdoms fell in 1974 TA, even the pretense of the existence of "Arnor" was dropped, and the last remnants of Arnor were abandoned.
Rohan didn't exist as such until the year 2510 of the Third Age. From that point on, it remained an independent kingdom.
Gondor, meanwhile, had fallen in stature, strength, and importance over the course of the Third Age, but it never ceased to exist.
The Reunited Kingdom and Rohan:
When Aragorn was crowned as King, Arnor popped back into existence (as half of the Reunited Kingdom) for the first time in a thousand years; until then, Rohan and Arnor had never existed at the same time. After Aragorn's ascension to the throne, Arnor and Gondor were united, but Rohan remained separate and independent from - though extremely friendly with, and closely allied to - both of them.
Rohan and Gondor:
Gondor had claimed "Calenardhon" - the land that would later become Rohan - as part of its own kingdom until 2510 of the Third Age, when Eorl of Rohan came to the aid of Gondor's forces in battle. As a reward for Rohan's actions, Cirion, the Steward of Gondor, granted Calenardhon to the Rohirrim.
‘Thus [Eorl of Rohan] came to the battle of the Field of Celebrant, for that was the name of the green land that lay between Silverlode and Limlight. There the northern army of Gondor was in peril. Defeated in the Wold and cut off from the south, it had been driven across the Limlight, and was then suddenly assailed by the Orc-host that pressed it towards the Anduin. All hope was lost when, unlooked for, the Riders came out of the North and broke upon the rear of the enemy. Then the fortunes of battle were reversed, and the enemy was driven with slaughter over Limlight. Eorl led his men in pursuit, and so great was the fear that went before the horsemen of the North that the invaders of the Wold were also thrown into panic, and the Riders hunted them over the plains of Calenardhon.’
The people of that region had become few since the Plague, and most of those that remained had been slaughtered by the savage Easterlings. Cirion [the Steward of Gondor], therefore, in reward for his aid, gave Calenardhon between Anduin and Isen to Eorl and his people; and they sent north for their wives and children and their goods and settled in that land. They named it anew the Mark of the Riders, and they called themselves the Eorlingas; but in Gondor their land was called Rohan, and its people the Rohirrim (that is, the Horse-lords). Thus Eorl became the ﬁrst King of the Mark, and he chose for his dwelling a green hill before the feet of the White Mountains that were the south-wall of his land. There the Rohirrim lived afterwards as free men under their own kings and laws, but in perpetual alliance with Gondor.
- The Lord of the Rings; The Return of the King, Appendices
Arnor and Gondor:
When Númenór was destroyed, the Faithful Númenóreans who had survived sailed to Middle-earth and established two separate but allied Númenórean kingdoms: Arnor in the north and Gondor in the south.
The last leaders of the Faithful, Elendil and his sons, escaped from the Downfall with nine ships, bearing a seedling of Nimloth, and the Seven Seeing-stones (gifts of the Eldar to their House); and they were borne on the wind of a great storm and cast upon the shores of Middle-earth. There they established in the North-west the Númenórean realms in exile, Arnor and Gondor. Elendil was the High King and dwelt in the North at Annúminas; and the rule in the South was committed to his sons, Isildur and Anárion. They founded there Osgiliath, between Minas Ithil and Minas Anor, not far from the conﬁnes of Mordor. For this good at least they believed had come out of ruin, that Sauron also had perished.
‘Eriador was of old the name of all the lands between the Misty Mountains and the Blue; in the South it was bounded by the Greyﬂood and the Glanduin that ﬂows into it above Tharbad.
‘At its greatest Arnor included all Eriador, except the regions beyond the Lune, and the lands east of Greyﬂood and Loudwater, in which lay Rivendell and Hollin. Beyond the Lune was Elvish country, green and quiet, where no Men went; but Dwarves dwelt, and still dwell, in the east side of the Blue Mountains, especially in those parts south of the Gulf of Lune, where they have mines that are still in use. For this reason they were accustomed to pass east along the Great Road, as they had done for long years before we came to the Shire. At the Grey Havens dwelt Círdan the Shipwright, and some say he dwells there still, until the Last Ship sets sail into the West. In the days of the Kings most of the High Elves that still lingered in Middle-earth dwelt with Círdan or in the seaward lands of Lindon. If any now remain they are few.’
The might of Hyarmendacil no enemy dared to contest during the remainder of his long reign. He was king for one hundred and thirty-four years, the longest reign but one of all the Line of Anárion. In his day Gondor reached the summit of its power. The realm then extended north to the ﬁeld of Celebrant and the southern eaves of Mirkwood; west to the Greyﬂood; east to the inland Sea of Rhûn; south to the River Harnen, and thence along the coast to the peninsula and haven of Umbar.
The Reunited Kingdom and Rohan:
In Éomer’s day in the Mark men had peace who wished for it, and the people increased both in the dales and the plains, and their horses multiplied. In Gondor the King Elessar now ruled, and in Arnor also. In all the lands of those realms of old he was king, save in Rohan only; for he renewed to Éomer the gift of Cirion, and Éomer took again the Oath of Eorl. Often he fulﬁlled it. For though Sauron had passed, the hatreds and evils that he bred had not died, and the King of the West had many enemies to subdue before the White Tree could grow in peace. And wherever King Elessar went with war King Éomer went with him; and beyond the Sea of Rhûn and on the far ﬁelds of the South the thunder of the cavalry of the Mark was heard, and the White Horse upon Green ﬂew in many winds until Éomer grew old.