Is it possible? The only reason I'm playing with the gog version is because me and my friend want to play through the domimatrix DLC together but I don't have the DLC on steam.

Also, can my save files be shared?

From the game card :

There does seem to be an explicitly titled "GOG Galaxy Matchmaking" button in the multiplayer menu, but can the LAN function be used to bypass that, so that a GOG and a Steam user use Evolve, Hamachi or Tunngle (or, in an ideal case, an actual LAN) to connect?