Marco,
There's not much stuff here.
I'm off for vacation to the islands tomorrow so unfortunately I will not be in the city to meet and give you more info face-to-face.
There is only one solid and true vendor in this country, name is WellShaved.gr (his name is George and speaks good English), he is supposedly shut down for holidays for all August but I know he'll be in the store to do housekeeping, so you may give him a call. His store is in the port of Piraeus, it is easy to find. Just let him know (call him) and I am sure he will not say no if you want to drop by.
He carries most common brands, his e-shop is something like Connaught Shaving / Shaving.ie, this style of product range, with slightly less stock of course as the market here does not allow for high volumes (as compared to the UK).
Downtown Athens and also in the countryside/mainland there are small shops which occasionally carry vintage shaving creams, after shaves, razors (especially injectors) but the few folks over at the Greek shaving forum who have found such shops in the small alleys of Athens have very seldom shared where they have found what. So there is stuff but you have to be adventurous and lucky to find.
There is the flea market in Monastiraki (downtown Athens) but what's there is already inflated in the price. There are other flea markets, e.g. the one next to metro station "Elaionas" can give good bargains but you should be there at 7am latest Sunday am and be unshaved, filthy dressed, etc.
Regards and enjoy your trip.
Konstantinos