In Ancient Rome, slaves addressed their masters as Dominus or Domina (male or female, respectively).
Would the slaves (or servants) of Ancient Greece have used a similar title, or would they have simply used their masters's names?
ἀνάξε (pronounced ah-NAHX-eh) is the vocative, if I've handled the accent right.
I vaguely suspect it might be ἄναξε (AH-nax-eh) - my greek is rusty.
Example (Odyssey 24.251):
οὐ μὲν ἀεργίης γε ἄναξ ἕνεκ᾽ οὔ σε κομίζει,
"It is not on account of your idleness your master does not take care of you"
First of all, your assumption that slaves in Rome addressed their master as "domine" is not true. The language used in the household was completely different than the "silver" Latin you read in Cicero or Seneca. Vernacular Latin had large amounts of Greek slang in it and the lower in the class the person, the more slangy it got.
Words like kurios and dominus are literary words that would be not normally be used in everyday speech by a slave. The normal word in both Latin and Greek was the Greek slang
If you read "Latin Forms of Address: From Plautus to Apuleius" by Eleanor Dickey it confirms my answer: slaves almost always call their master/mistress erus/era to their face. Note that "ere" is the vocative form, so that is the form that would normally be used (the Greek equivalent is the same). You will sometimes find the word in Latin dictionaries under "herus". Don't be fooled by some of these dictionaries into thinking this is an "official" Latin word, it's actually very colloquial Greek.