PM just sent on my own experiences with this process, in 2016, from Switzerland (non-EU country). Since this was one of my personal belongings when I moved from Switzerland to France, i did not have to pay any import taxes. (subject to a requirement not to sell the car within a one-year period).
I do think not having a registration will cause problems. When I got my certficate from MB France (by the way, that cost me EUR 150, and they did not require proof of a KPH speedometer, although the car was originally, way back, imported from the US) they did require a copy of the Swiss registration, front and back. So I hope Babis does have the Greek registration document. If not, it should be possible to obtain a copy.