You can't start a religion (sort of) in another holy city.

Havent really had the chance/wanted to try this out before, but in case anyone was curious - I had the chance for an experiment. I am playing Immortal Carthage and Tilted Axis - Ethiopia managed to found a religion extremely quickly and the main continent was near the island I was stuck on - so I planned to snipe his capital city and hoped to used my great prophet on his former capital (and his holy city) for a better religious focal point. Well I sniped his capital - and sent in my great prophet to see if I could found another religion in another holy city
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The result?

I could, but... none of the population was converted to my new religion leaving my empire with no cities that followed my religion, but left the holy city as holy city for 2 different religions. I didn't keep playing from that point to see which religious influence would eventually take over (his holy city influence or mine) - but I thought that was interesting.

I sacked the Celts Holy City and was able to use my GP to spread religion, which converted all the citizens but didn't change the Holy City. I'm guessing you could found your religion, use your first Missionary to spread religion, then park an inquisitor to keep the other religion from taking root again. I still think it would exert influence for bot religions though.

Whether that's a good idea tactically is debatable, but logistically I think you can do it.

^ Just "parking" an inquisitor doesn't stop the pressure from other cities, it just stops AI from using their Great Prophets to your city. Unless you mean to use it when your religion gets overruled.