Even if you uppercase something you still need to support your claim with some proof, because otherwise one can come up with a refutation similar in it's nature, like this one:
No, it DOES have something to do with Ashkenazi.
See, it's not that convincing per se, so I'll provide some explanation.
Jews have a long-documented history of adopting same name patterns as the community they are leaving in. That's why we have a lot of German-like (Tittenstein, for instance), Slavic-like (for instance, Rabinovich) and Luthuanian-like (as Margoulis) last names.
Jews are no strangers for Caucasus in general and for Georgia in particular.
It's actually quite complicated because, generally speaking, Georgian jews in the majority are not Ashkenazis, but still, there were Georgian Ashkenazis as well.
Some actual people with this last name are actually of a Jewish origin, like, for instance:
- Liya Efimovna Ashkinadze (bio in Russian)
- Esther Ashkinadze (link in Russian/German)
- Grigory Ashkinadze (link in Russian)
Apart from that (but this should be investigated) the -dze ending can be just mutated ending of si/zi for jews of non-Georgian origin.