I've seen phrases like Как его зовут? in several places, but when I look up зовут, it appears to be the third person plural. Why is it being used for the singular, here?
Зовут is really the third person plural, "[they] call", it is an impersonal verb and has no explicit subject in the sentences about names, so
Как его зовут? can be translated word for word as "How him [they] call?" that is "How do they call him?" And the answer is "Его зовут Алексей." - "Him [they] call Alexey." - "They call him Alexey."
That is how these phrases about names are built in Russian.
"Его зовут X" is similar to "They call him X". That's why it is always 3rd person plural for everyone. And that's why it you use Accusative case for the one whose name you tell.
In Russian this structure is one of the very popular ways to make a pseudo-passive. If an action is performed by assumed "unstated persons" — then you can express "it is done" by saying that "(they) do it" while omitting "they":
На углу строят здание = A building is being built on the corner.
Так не говорят. ~ People don't talk like that (= "sounds unnatural/bad")
Я думаю, нам помогут = I think we'll receive help.
Needless to say that "Меня зовут Джон" is a construction that crystallized this way and a native hardly even notices "passive" here.