We're releasing everyone tomorrow.
We need the name of the person or persons you want to authorize to pick you up.
I, uh, I don't really have anyone.
I'm kind of between "Persons" right now.
In this case, "between persons" is a facetious play on words which is based on the phrase "between jobs" (meaning 'unemployed').
The term 'unemployed' can have negative implications and people often avoid using it when talking about their own situation. At some point, it became fashionable to use the phrase "between jobs" as a euphemism for being unemployed. In your example, the speaker is attempting to do the same thing by using the euphemism "between persons" as a way to avoid directly saying that they essentially have no friends or loved ones who can pick them up.
My guess is that the writer of the first sentence really wanted to say something more specific (depending on context) such as 'wife', or 'boyfriend', but realised that that would be making unreasonable assumptions. The speaker has noticed that ,and is teasingly using the word person rather than say something like "I am between girlfriends right now."– JeremyCNov 27, 2018 at 23:02