First, just to answer your original question as best I can, some Stadia games started out as PC ports, and some started out as console ports. It really depends on the game and the developer. The Stad...
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First, just to answer your original question as best I can, some Stadia games started out as PC ports, and some started out as console ports. It really depends on the game and the developer. The Stadia stack looks more like a Playstation in some ways (UNIX-like operating system with a modern 3D graphics stack on top), but since most games already have to target multiple platforms already, they typically have a portability layer of some kind, so it may be easier to start with a PC port in some cases, even though that is a Windows operating system with a different modern 3D graphics stack on top.
As for cross play, @DaBaldEagul is right – it's certainly possible technically. It just depends on the game and the developer. Often it comes down to whether the particular game version is synchronized across platforms or not. If they are synchronized releases, then having all of the platform ports be on the same version makes cross-platform play a lot easier, since the message formats are all compatible and the behavior on each end will be the same. So, the game servers know all of the clients will behave properly even if they are all different platforms. But, if the platform port versions aren't synchronized, the developer may just decide it's too much effort to avoid making a breaking change to the message formats and ensure the behavior remains "compatible" across all of the currently-released versions. In that case, cross-play is unlikely to be supported. There's a sort of "hybrid" state, though, where you might see a Stadia port (for example) that started out life as a PC port end up supporting cross-play only with the PC version, but not the other console versions. So, in that case, yes – the port type may be relevant for cross-play.
This is all assuming the developer has put any effort to supporting cross play at all in the first place – that requires building a portability layer for things like player identity, friend list, in-game chat/audio, game-saving, etc. Game saving across platforms (cross-progression) may be more likely to be supported than actual cross-play. So, yeah – modern game development is complex, and things like cross-progression and cross-play require effort. Each developer has to decide how to invest the limited engineering resources they have and make a trade-off between putting more into the game itself vs supporting these cross-X features.