I remember the Dominions video game series being based on a PnP RPG, but I can't seem to find the rpg anywhere (or the things indicate that there was one, besides the nature and structure of the games themselves). What is the Dominions RPG and where can I get it?
The game you're looking for was inspired from the author's Ars Magica game. With sufficient effort in reading the Realms of Power: The Divine, a sufficient approximation of the domain/realm interactions can be intuited.
With that said, looking at the spell list, I can state with some certainty that the mechanical heritage between Ars and this game is... remote, even as the discussion on page 3 talks about one of the primary thematic conflicts in Mythic Europe. On p.125 the author speaks of their "Ars Magica campaign." but it's probably worth differentiating "inspired by concepts in" versus "based on." And the dominion mechanics they cite on p.104 ... would be difficult to map to Ars in any 1:1 way, though as a GM, I "can see what they're getting at."
It could be Ars Magica.
The main manual PDF of dominions 4 (6,8 MB) mentiones several inspirations in it's A short history of Dominions Chapter. The only RPG mentioned there is Ars Magica.
Other non-PnP inspirations mentioned are: