Does Tatarewicz derive from a tribe of mountain Jews/Tartars?
I don't think so.
The name looks like a patronymic - meaning a son of Tatar - from
Tatar which was either a ethnonym (Tartar) of the forefather or his moniker. If he had even lightly slanted eyes or mongoloid forehead he could obviously be called Tatar. There are 184 persons in Poland with this surname:
How and when were surnames officially/legally assigned in old Poland?
Surnames in old Poland were not officially assigned. They were recognized.
If somebody wanted to write a record about his possession or declare an inheritance, he went to the
gród (court) and made it written to the records. He declared himself by surname he used to be called.
Next: if he achieved some privilege from the king - he was addressed by the surname. Sometimes, especially if he was of the foreign origin, he was granted with a coat of arms and surname. Later if he wanted to prove his nobility, he produced all such evidences.
Assignments started in police states which occupied Polish territory after partitions.