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

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

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.