No, P.K. doesn't mean anything other than it's a common first name-second name combination in India, with K many times standing for Kumar, a very common middle name/ surname in India.
However, peena in Hindi means to drink, and kay here would mean after, having, which makes Peekay as 'having drunk' which sounds interesting.
That's also the reason you can see people who are fondly called Peekay by friends instead of their actual names being called, because it's not derogatory, but still funny in a way.
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