I was looking into getting Relic to play with some friends but despite the game supporting up to 4 players I didn't see much interaction between players in the video. Is it literally just a race or can you interact with other players in any way?

sadly for that you need the expansion nemesis, with nemesis you have 2 pvp systems

Even with Nemesis, PvP is lackluster and limited. "Good" characters can still barely do anything about each other, it'll mostly be Nemesis character vs other characters (and if you have 2 Nemesis, they can't even attack each other, come on!). Since there is no elimination (having to start over fresh) in Relic, that too takes out some of the point from going after other people, worst of all, can't take Relics from people who have them, really making PvP all but moot. If you want PvP mayhem and glory in a similar vein, Talisman is the game. Relic might not be a poster-boy for multiplayer-solitaire, but it isn't far off.

Well, the Relic expansion offers those apostate items, which are more powerful than the mundane ones. Since we introduced this expansion, there's a lot of PvP going on in our games. Even more, if there's also a Nemesis on the board