Civilization – If you conquer a holy city do you take “ownership” of that religion

All cities remember only the original civ that built them. It's much more cost-efficient in the long run to raze cities and resettle in the same place with your own settler given the oftentime nigh untenable cost of the courthouse building (and the fact that it can't be purchased).

One thing to keep in mind is that any major religion is superior to no religion at all. Yes, it gives a bonus to your opponent, but it gives a larger bonus to your city. Therefore, its advisable to use these GPs to spread religion to cities that do not have, and are unlikely to get, Shinto. Island colonies far from other cities are the best example. They would not otherwise get a religion, so it can be wise to spread buddhism or confucianism there.

Finally, look at the benefits provided by those religions. If you have a specialty city, it may be useful to add a specialty religion. For example, if you have a pure production city, and Buddhism gives the 15 percent production bonus, it may be wise to spread Buddhism there in order to get said bonus.