Does a European passport get an exit stamp at the time of leaving Europe for Azerbaijan or Georgia? Also will the destination countries of Georgia and Azerbaijan put an entry stamp once you enter there?
If you mean leaving Schengen Area for Azerbaijan or Georgia, then, shortly:
If you're not a citizen of EU or EDTA, then your passport will be stamped, because Azerbaijan and Georgia aren't in the Schengen Area.
For a longer answer, check Schengen Area section of Wikipedia article about passport stamps.
Passports of EU/Schengen states, Andorra, Monaco and San Marino are never stamped on leaving an EU/Schengen country, unless possibly if you ask for it.
Azerbaijan and Georgia do stamp all foreign passports (with Georgia also stamping their own passports on exit). If, for some reason, an EU citizen wants to avoid Georgian stamps, they can use a national ID card.
If you ask for a stamp, you may get one. Lithuanian border guards are really friendly and they stamped my Lithuanian passport in Kaunas (flying to UK) and in Vilnius (to Ukraine) and returning. However, Latvian officer on Russian border at Grebņova declined. British border guard at Luton also refused, so it depends on officer's mood.
Schengen border code says that Schengen/EU passports are not stamped, but the actual stamp cannot make illegal consequences for an EU/EFTA citizen, as they have a right to be in EU/EFTA for as long as they want.