I want to change and update shower stall fixtures. I now have Delta. Do I need to replace with Delta fixtures? I do not want to disturb the tile.

I also want to replace fixtures on a roman tub. Do I need to replace with Delta also?

Shower: you should be able to purchase new trim for your Delta faucet. However, if you have Delta shower valve inside the wall - you have to stick with Delta brand trims.

Roman Tub: you should have at least one access door somewhere behind the tub. That access is there to allow you to work on your faucet. Assuming you have deck-mounted faucet, you can remove entire rough through that opening and reinstall new one. The new Roman Tub faucet then can be any brand. That applies to the spout, too.