You should specify what you mean by a polygon: a broken line or a surface.
If we are talking about a broken line, then some classification is given here:

 MR1703691 Kapovich, Michael; Millson, John J. On the moduli space of a spherical polygonal linkage. Canad. Math. Bull. 42 (1999), no. 3, 307–320.

If we are talking about surfaces, this is a different problem which is much more complicated.
When the angles are sufficiently small, so that the whole polygon is a subset of the sphere,
a classification is obtained in

F. Luo and  G. Tian, Liouville equation and spherical convex polytopes, Proc. AMS, 116 (1992) 4, 1119-1129.

If the interior angles can be arbitrarily large, there is no known classification, even for the case of quadrilaterals. Some partial results can be found in 

Eremenko, Gabrielov, Tarasov,   arXiv:1405.1738 
Metrics with conic singularities and spherical polygons

We actually have a classification of quadrilaterals up to isometry but it is not ready for publication yet.

To answer your more specific question. A spherical polygon generally depends on 2n-3 real
parameters: $n$ angles and $n-3$ additional parameters. For these additional parameters,
one can take certain conformal moduli, for example accessory parameters in the differential equation which is associated to this polygon. So for $n\geq 4$, we have 5 parameters, but whether there exist two quadrilaterals with the same angles and perimeter we did not
investigate. For $n\geq 5$ polygons with the same angles and perimeter certainly exist,
by dimension count.

EDIT. That $n$-gons with the same angles and same perimeters exist, can be obtained from our other paper,  arXiv:math/0405196 where polygons whose angles are integer multiples of $2\pi$ are classified.