If look at the Latitude lines - blue circles below - and pick one, each point $P$ not in one selected circle/line, will only lie in one other blue circle, circle here taken as line, and will follow the exact definition of postulate 5.
So we have also the same notion of a parallel line.
Is this incorrect, and why ?
See the blue lines I imagine just below:
It appears that your proposed model geometry is the unit sphere in ${\mathbb R}^3$ and "lines" in this geometry are the longitudes. I also suppose that the north and south poles also qualify as "lines" in this geometry (otherwise, Playfaire's Axiom will be violated). Such a geometry indeed will satisfy Playfaire's Axiom. However, it will violate other axioms/postulates of Euclidean geometry, e.g. the 1st postulate. (Arguably, it will also violate the 2nd postulate and, depending on your interpretation, the 3rd one as well. Even the 4th one is suspect since it is very unclear what the "right angle" means in this geometry, where there are no transversal lines and where some points are also lines!) By the way, I prefer Hilbert's axioms since they provide for a clean and complete treatment of the subject.