That there is no such line in spherical geometry is not part of Playfair's axiom and, as you point out, is false. If you want to clearly differentiate between Euclidean and spherical geometry you have to reword the axiom, for instance,
For every line and point not on the line, there exists exactly one line passing though that point and parallel to the given line.
There is something else going on here: spherical geometry obeys Playfair's axiom, as you've written it. But it obeys the parallel postulate as well!
This raises a pressing question: which of Euclid's axioms is violated by spherical geometry? For instance proposition 27 is false in spherical geometry. It may not be obvious at a glance from the classical wording of the five postulates.
The key thing to understand is that Euclid's arguments, though tremendously influential, are not rigorous or complete by modern standards. If you follow Euclid's arguments very carefully you will find propositions where his arguments break down in spherical geometry, because of unproven "obvious" assertions that do not appear in his list of axioms. For instance proposition 16, a key step in Euclid's treatment of parallel lines, makes a claim about the relative magnitude of angles that is true in the plane but not on the sphere.