How to find fundamental cycles in a multiple edge graph?

I tried to find the fundamental cycles of a multiple edge graph below, but it doesn't seem to work. Is it true that the built-in function FindFundamentalCycles does not work with multiple edge graphs? Or did I make a mistake or not understand the function?

g = Graph[{1 <-> 2, 2 <-> 3, 3 <-> 2, 1 <-> 3},
VertexLabels -> "Name"];
cycle = FindFundamentalCycles[g];
HighlightGraph[g, #] & /@ cycle


Version: 12.1.1 for Microsoft Windows (64-bit)

A workaround for highlighting a multigraph from @HenrikSchumacher answer.

g = EdgeTaggedGraph[{1 <-> 2, 2 <-> 3, 3 <-> 2, 1 <-> 3},
VertexLabels -> "Name"];
cycle = FindFundamentalCycles[g];
HighlightGraph[g, EdgeList[g][[EdgeIndex[g, #] & /@ #]]] & /@ cycle


Mathematica's implementation of multigraphs is called EdgeTaggedGraph. So better use the following

g = EdgeTaggedGraph[{1 <-> 2, 2 <-> 3, 3 <-> 2, 1 <-> 3},
VertexLabels -> "Name"];
cycle = FindFundamentalCycles[g]

{
{UndirectedEdge[3,1,1],UndirectedEdge[1,2,1],UndirectedEdge[2,3,1]},
{UndirectedEdge[3,1,1],UndirectedEdge[1,2,1],UndirectedEdge[2,3,2]}
}


However, HighlightGraph does not seem to be able to highlight these cycles correct. (At least not in version 13.3 that I am using at the moment.)