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.