As many have already said, a game is a shooter if you shoot something.
Nothing fancy, just that the game has shooting as a core gameplay element.
As for the idea of what is a "proper" shooter (as silly as the whole thing)... well, this is probably easier to answer than one might think:
Is the primary if not sole intent of the game shooting things to make them fall over dead?
Especially if the game has a strong focus on multiplayer (particularly of the online variety), where the vast majority of gameplay boils down to shooting the opposition? That's probably closer to the idea of the so-called "proper shooter". Shoot, shoot, shoot! Kill, kill, kill! It's fairly absurd, but I suspect that this is what's going on here.
Of course, many other games use shooting elements... but aren't really something that could be called a shooter. The Mass Effect series uses third-person shooter mechanics for the combat, but it also has many other elements which could be called more important: character progression/customization, dialogue tress, character development, and storyline. For another perspective, there's also the Metroid Prime sub-series; very strong focus on puzzles and exploration, significantly moreso than the first-person-shooter-based combat (most of the bosses are puzzles rather than grudge matches).
Both series of games are technically shooters, but neither have shooting as their primary element... because they aren't just about combat. Their combat is shooter-based, but they have more to it than that.
Back to the absurdity of a "proper shooter"... it would seem the friend of the topic creator seems to be of the opinion that the entire point of a shooter (hence the "proper" label) is just shoot things and nothing more. A self-reinenforcing narrow mindset, really.