This is a partial answer, with a few assumptions.

Black will win with perfect play Each board is either winning or losing, a board is losing if playing first on it would mean that your opponent would win. Regardless of position.

Thus:

There starting configuration of boards are [][][][][L] Note these boards are in no particular order and rely soely on the hint Thus white can go on board 5 so black will reply on 5 and thus it will be presented with the same position so white must change the state to [L][][][][L] black goes on another board [L][L][][][L] and white must again play on a winning board or play till they can't on a losing board and so eventually white will play [L][L][L][][L] then black completes the set to [L][L][L][L][L] and so whenever white goes black will go on the same board until black has won

Still to be shown:

whether the starting assumption is true, the [L] board and the optimal strategy for each board

This is a partial answer, with a few assumptions.

Black will win with perfect play. Each board is either winning or losing. A board is losing if playing first on it would mean that your opponent would win. Regardless of position.

Thus:

The starting configurations of the boards are [][][][][L] Note these boards are in no particular order and rely solely on the hint Thus white can go on board 5, so black will reply on 5 and thus it will be presented with the same position. So white must change the state to [L][][][][L]. Black goes on another board [L][L][][][L] and white must again play on a winning board or play till they can't on a losing board. So eventually white will play [L][L][L][][L]. Then black completes the set to [L][L][L][L][L]. So whenever white goes, black will go on the same board until black has won.

Still to be shown:

whether the starting assumption is true, the [L] board and the optimal strategy for each board