Like a lot of Ancient Greek verbs, aphiēmi has an impossibly broad range of meaning. Literally, it means “send from”. If you look at the range of meanings in LSJ (which is Classical Greek rather than Biblical Greek, but that helps us avoid the temptation of theologically influenced glosses), you’ll find:
I. send forth
II. send away
II.1.b. let go a person, release a person; and as a ditransitve, release a person from something, acquit someone from something.
As others have said, releasing someone from a debt is the same thing as forgiving someone’s debt. (In fact, one of the instances of “release a person” given is from Polybius, “release them untaxed”, i.e. let them not pay their taxes.) But the concept involved is forgiving a debt (or “trespass”), not forgiving in general.