What is the German / European equivalent of FAA Order 7110.65?
The „handbook“ for ATCOs in Germany (!) is the MO-ATS (Manual of Operations for Air Traffic Services) which is an internal DFS (Deutsche Flugsicherung - the German ANSP) document and therefore not available to public.

However it is based on the German AIP and ICAO DOC 4444 which are both available to public. For the international legislation check Milo Morkus’ answer.

In the German AIP phraseology examples are found in AIP ED GEN 3.4 Communication Services