Hamman asked: Would you like some company? Do you want some company? Is it okay if I join you? Do you mind if I join you? In general, 'care for some company' means all of these or any combination. It depends on the context. Neo responded: Yes, I would. Yes, I do. Yes, it is. No, I don't mind. It's okay if you join me. I don't hear the phrase everyday, but it's not uncommon. If you said it, it wouldn't sound strange; it's a native-sounding phrase. https://www.collinsdictionary.com/us/dictionary/english/care: 4. TRANSITIVE VERB/INTRANSITIVE VERB [no cont] You can ask someone if they would care for something or if they would care to do something as a polite way of asking if they would like to have or do something. [politeness] Would you care for some orange juice? I hope this helps.