Can you transfer money on Memorial Day if you share the same bank account?

If your bank allows you to transfer money, then you can transfer money. If they're closed (and most banks are closed on Memorial Day), then it's going to be difficult, but you may be able to do it at an ATM or online. It's not clear what you mean by "if you share the same bank account". Transferring money from one account to the same account is kind of a null operation.