svůj means "one's own," so when referring to the first person conjugation, it means "my," second-person means "your," third-person means "his, her, its," etc.

On pijí svůj čaj, "He's drinking his tea," for example.

Yes, "my" is a good translation here. How else would you translate into idiomatic English?

"I am drinking own tea?" I don't think so! Sounds like a parody of a Russian, Czech or Pole in a nightclub act.

There is no difference in this sentence. But when you say "Vezmi si svůj čaj, ne můj" (Take your tea, not mine) then, there is clear difference.

Anyway we would (almost) never say "piju svůj/můj čaj". We say mostly only "piju čaj."

svůj means "one's own," in this case "my own."