Do function generators produce AC signals in the sense we know?
An AC signal is a signal that Alternates, which means it changes direction over time.

Even the cheap DIY "function generators" can generate such a signal (if I apply that signal to a capacitor, the capacitor will charge and discharge so  the current changes direction) so yes, that's AC.

Indeed some might not be able to produce a negative voltage and can only output the AC signal superimposed on a DC signal. You could use a capacitor to block that DC voltage.
You could use a DC voltage source as a reference such that it compensates for the DC voltage.

Making a "proper" balanced output signal with no DC present requires a slightly more complex circuit. For a DIY function generator having that DC voltage present might actually not be an issue.