First of all, none of the good ones are based on arbitrarily chosen axioms. The good ones are based on axioms that are known to have useful interpretations and therefore the opposite of arbitrarily chosen.
A logical system certainly can have arbitrarily chosen axioms; but, if so, it is just an exercise in logic-chopping.
As for what it means to say that a logical system can have arbitrarily chosen axioms—-all it means is that, since logic is about what follows from what, it doesn’t technically matter what one’s starting points are, only that one’s deductions therefrom are valid.
But, of course, it does matter what one’s starting points are, since, when they’re ill-chosen, the resulting system is completely devoid of practical or even theoretical interest.