Tl;dr: Go ahead and mark the cards - the difference in gameplay is real but negligible.
Here is one of the rare scenarios where it makes a difference, and even then it's a tiny one: Your opponent knows you are holding card X but does not know your whole hand - in particular, they don't know that you're holding a second copy of X.¹ Then they force a discard (e.g. with Imperial Frigate) and you discard an X. Now if the two copies of X are indistinguishable, your opponent can't tell you're holding another one - for all they know you discarded the one they knew about. But if one is marked then you have to be careful: if you discarded the X they didn't know about, they can tell that you're still holding the other one and can make the remaining decisions of their turn with that information in mind.
In Star Realms this issue is small enough that you should ignore it, but e.g. in pro Magic, similar considerations make it commonplace for players to avoid using multiple versions of the same card.
¹This can happen if they return a base to your hand (e.g. with Mega Mech), or if they can remember all your cards so that when you draw the last couple, shuffle, and draw a few more, they know the pre-shuffle draws by process of elimination.