I don't think it was necessary for a Slytherin to be a pure blood.That was just another Slytherin trait,like pride, ambition,cunning etc.Not all Slytherins can have all these traits.That also happens in the rest of the houses.Voldemort was a half blood,but he became a Slytherin as well.
Slytherins are, according to the Sorting Hat, devious, secretive, cunning, and ambitious. The blood purity thing was not started at Hogwarts in itself, so far as I know. Quite possibly, when the hat saw Snape's mind, it could see his secrecy and cunning. The Sorting Hat can't tell if you're a pure-blold, half-blood, या Muggle-born.
It isn't because you're a half-blood या muggleborn that आप can't be in slytherin. The sorting hat doesn't take account of the *purity* of your blood when he is choosing which house आप will most fit in, but your values. If आप are ambitious=slytherin, brave=gryffondor, hufflepuff=loyal, ravenclaw=clever. There are some muggleborns in slytherin but because of the majority of slytherins that have prejudice against them they tend to have low profile, to not be noticed. Because the don't want to be bullied. So please no. Just because you're a slytherin doesn't mean आप can't be a muggleborn या half-blood.
SALAZAR Slytherin was pro pure blood. (although there is some evidence that he is और tolerant towards half-bloods being that the basilisk went after muggleborns. But that could just be Voldemorts view).
Regardless, Hogwarts itself (including the Houses and the sorting hat) does not base anything off of "blood" status. Students are sorted based on their traits. And quite frankly, Snape is very "Slytherin" (sly, cunning).