There are 5 axioms that define euclidean space and I believe that all hold also for a sphere. The definition of axioms from wikipedia:
- "To draw a straight line from any point to any point."
Definition of "straight" might be tricky, but I think that it's simply a shortest path on the surface.
- "To produce [extend] a finite straight line continuously in a straight line."
Hmm... "Continuosly". Does it mean that I can't draw the extension over the original line? I'm not sure about that, but the axiom doesn't forbid that.
- "To describe a circle with any centre and distance [radius]."
If circle is defined as a set of points with the same distance from the center, no problem there.
- "That all right angles are equal to one another."
I don't know what to say about this one, but I think it's true.
- The parallel postulate: "That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles."
All straight lines on the sphere intersect, so this is a no-brainer. I see no if and only if.
So where is the problem? Which axioms don't work on a sphere? Is there a problem with basic words like "line" and "straight"?
The basic premise of your question is incorrect: the five axioms you listed do not characterize the Euclidean plane (and in any case need to be formulated a lot more precisely for that question to even make sense). They are the axioms Euclid listed, but actually he implicitly assumed several other axioms.
In particular, one concept that is crucial to Euclidean geometry is order: given three points $A$, $B$, and $C$ on a line, we can say one of them is between the other two and this notion of "betweenness" satisfies a certain list of axioms (see Hilbert's axioms, for instance). There is no appropriate notion of "betweenness" on the sphere. Intuitively, since a "line" on the sphere is a great circle, you can't say one point is between the other two because which points are between $A$ and $B$ depends on which side of the circle you use to travel from $A$ to $B$.