Yeah they aren't a unit in the sense that they make a non-quantifiable measure quantifiable. In dimensional analysis they would have the dimension [1]. But they can still be regarded as a unit since they act in the exact same way, just like other factors do. But yeah, they are more akin to the SI prefixes like kilo, or something like a dozen or a gross.

A mole isn't just convenient. I'm sure there's a youtube somewhere that explains it but advagadros number is a product of the fact that: 1) every atom of an element has a weight (or at least an average) and 2) atoms interact in integer quantities. If you put those two together there is a common multiplier for a stochiometric equation that is related to the mass of a given atom in that stoichiometry. That multiplier is the the mole.

edit: I guess that's kind of a factor, but it's really more of an derived unit. If there was a new element discovered a mole would still describe it's stoichiometry, and IIRC that's how a lot of the periodic table was filled out my Mendelov.