...not that I have much experience with, nor interest in this sort of scenario (I'm in this skill for the sake of completeness), but to me "may I offer you a drink?" in English just does not sound ...flirty. It makes me think of an upscale bartender or a butler or waitstaff in a fancy restaurant or someone playing host/ess. Granted, the last could occur within a romantic context, but that does not seem to be the usual caliber of Duolingo's flirting unit.
Is this sentence any more casual in Spanish?
because te in this case means to you if it were "I offered you a drink" it would be "te ofrecí una copa". or "I offered to you a drink". In spanish if the verb is in the infinitive, usually in a verb phrase, we attach it to the end. Lo siento si no está claro todavía, no puedo enseñar.
My english is not very good, but I will try explain you a little. The word "copa" in this sentence means literally "wine glass". Therefore a more aproximately meaning of the sentence can be: "Can I offer you an alcoholic drink?" for that reason this phrase can be used only in night clubs, bars or restaurants.
Uhh I don't know very well actually. I speak as a first language Spanish, I have never thought of that. Maybe using "te" at the end of the word is when you talk directly with someone, like: "puedes ponerte tus zapatos?" (Trad: May you put on your shoes?). But when it's "someone" (you are not talking directly with that person) you use with "se". Like: "Ella va a ponerse a estudiar a las 6:00" (Trad: She is going to study at 6:00)
That has two conjugated verbs. Sort of like: (Can I) (you offer) a drink. And the "you" in that case is a subject "you" ("tu"), not an object "you" ("te"). In Spanish, we use the infinitive for the second one.
In English, we use the infinitive form without the "to" (Can I give him; Can you give him; Can she give him, Can we give him; Can I be that; Can you be that; Can he be that; Can we be that; etc.)
Note: In this case, the "te" (direct object "you" for the verb "offer") is appended to the infinitive, "ofrecer". It can also go in front of the conjugated verb.