The following sentence is meaningless to me. What is the meaning of "buy" here?
I don't buy this person answering your mobile.
It sounds like "trust" to me, but I have never seen this meaning of "buy".
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In casual or informal English we can say we don't 'buy' (accept, agree to) an idea if we mean that we don't believe that it is likely or probable.
In the question, the speaker does not accept the idea of 'this person' answering the other person's mobile (cell) phone.
If you tell your teacher that a dog ate your homework, it is quite possible that he or she will not buy that (i.e. buy your explanation).
buy verb (BELIEVE) [ T ]
to believe that something is true:
She'll never buy that story about you getting lost!
"Buy" in this sense means "Believe" or more broadly, "accept" or "support."
"I don't buy that she was sick" -- she told people she was sick, but I don't believe it.
It can also be phrased as "buy into". According to the Cambridge Dictionary: "If you buy into an idea or plan, you give it your support or agree with it."
EDIT: I'm leaving this answer up in case it's something people can learn from, but it is incorrect. As people in the comments have noted, I interpreted "this person answering your mobile" as a noun phrase, but that probably isn't what the question-asker meant. "Buy" in this case can have its usual meaning of "believe that X (that this person is answering your mobile) is true).
As a native speaker (US), I've never seen "buy" used that way either, but I would make the same guess as you: That it means "trust". To me, you can only use that type of "buy" to mean "trust what the person is saying." So it is odd to say "buy [the person]" rather than to say something like "buy their story" / "buy their spiel", etc.
However, given that we can use "believe" in both contexts ("I don't believe his story." "I don't believe him.") it doesn't seem that crazy to me if some speakers have started using "buy" in both contexts too.