You can play all cards.
Each ally ability can be used (once per ability) if at least one other card of the same faction is in play (including bases), or if you have a base with special ability that allows triggering of all ally abilities.
So if you have 3 cards of the same faction in play, you may have upto 3 ally abilities to use. (I don't think there are cards with multiple ally abilities.)
There is in Star Realms: United. Multi-faction ships sometimes have an ally ability for each of the factions they belong to.