Why are some of the isomorphisms not bijective?

This is a bug. The length of each association returned should be VertexCount[g1], but it isn't.

Length /@ FindGraphIsomorphism[g1, g2, All]
(* {10, 6, 8, 6, 8, 6, 9, 9, 9, 9, 6, 8, 8, 10, 8, 10, 10, 9, \
9, 10, 9, 10, 9, 10, 10, 9, 9, 10, 8, 10, 10, 9, 10, 8, 9, 10, 9, 10, \
9, 10, 10, 9, 8, 10, 9, 10, 8, 10, 10, 10, 10, 9, 8, 9, 10, 10, 6, 9, \
10, 9, 8, 9, 6, 9, 10, 10, 9, 10, 10, 9, 10, 10, 10, 10, 9, 10, 10, \
9, 6, 9, 10, 10, 8, 9, 9, 6, 9, 8, 9, 10, 10, 10, 10, 10, 10, 9, 10, \
10, 9, 10, 9, 8, 10, 10, 9, 6, 9, 10, 10, 9, 10, 10, 10, 10, 9, 8, 9, \
10, 9, 6} *)

The bug is present in M10.4.1 but not in M10.3.1.

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As a workaround, try IGraph/M, which has superior isomorphism functionality.

To find all isomorphisms, you can use the VF2 and LAD algorithms in the package.

<<IGraphM`

IGVF2FindIsomorphisms[g1, g2]

IGLADFindSubisomorphisms[g1, g2]

The Bliss algorithm can find one isomorphism:

IGBlissGetIsomorphism[g1, g2]

Or get the number of isomoprhisms:

IGBlissAutomorphismCount[g1]
(* 120 *)

Or get the automorphism group, which can then be used to generate all isomorphisms:

group = PermutationGroup[IGBlissAutomorphismGroup[g2]]

iso = First@IGBlissGetIsomorphism[g1, g2]

AssociationThread[Keys[iso], #] & /@ (Permute[Values[iso], #] &) /@ GroupElements[group]

I think this is a bug of FindGraphIsomorphism.And you should give a report to Wolfram.This is what my thinking about it.

IsomorphicGraphQ[VertexReplace[g1, Normal[#]], g1] & /@ isos // Counts

<|True -> 64, False -> 56|>

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But I still think there are work-around to find the isomorphic graph.This is my solution.

Find the permutation group all of isomorphism

permute = GraphAutomorphismGroup[g1]

PermutationGroup[{Cycles[{{2, 4}, {6, 7}, {8, 10}}], Cycles[{{2, 4,3}, {5, 6, 10, 9, 8, 7}}], Cycles[{{1, 2, 8, 7, 5, 3}, {4, 6,9}}]}]

Then check what we get

ruleYouWant = Rule @@@ Transpose[{VertexList[g1], #}] & /@ 
   Permute[VertexList[g1], permute];
IsomorphicGraphQ[VertexReplace[g1, #], g1] &/@ruleYouWant

{True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True,True}