If you conquer a holy city do you take "ownership" of that religion?
Solution 1:
No. Even if you wipe out the civ, it is still not your religion.
The holy city (and other cities) will continue to spread that religion and you will gain the benefits of any pantheon or follower beliefs in it but it is still not your religion. You will not be able to benefit from its founder beliefs or add to that religion through enhancing or reformation.
If you buy missionaries or great prophets from that city, they will be of and will spread that religion.