I was able to get two FEN strings, which represent the same board, with the same full move counter, but different halfmove counters:
rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQkq - 0 1rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQkq - 1 1
which I think is not correct.
I was able to get two FEN strings, which represent the same board, with the same full move counter, but different halfmove counters:
rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQkq - 0 1
rnbqkbnr/pppppppp/8/8/4P3/8/PPPP1PPP/RNBQKBNR b KQkq - 1 1
which I think is not correct.