They do have the same root indeed, actually "πουλί" is a returned loan (I'm not sure that's the proper terminology tbh) from latin "pullus" that used to mean chicken, but also little/young bird, and young animal as well, which in turn is a loan from the ancient greek "πῶλος" which used to be a foal, a young horse ("πουλάρι" in modern greek, same root as well). Also from the same root is the word "πουλάδα" which is a young hen, but it's not a word you'll often come across, it's almost idiomatic.
About your other question, "πουλί" is indeed a euphemism for penis (I usually translate it to cock, since both words have the birds and fowl correlation :P ) but it's not one of the most vulgar slang words of those euphemisms. With the same root as well is "πούλος" (harsher slang) and "πουλάκι" (not a vulgar euphemism, kind of equivalent to "weenie")
I was referring to your reply to macaubert's post. Anyhow, I find it funny that people across the planet with a lot of cultural and historical differences and until recently without the opportunities of communication between them that the recent technological advances have given them nowadays, many many years ago found the same inspiration from the feathery animal kingdom to name their body parts :)