Accidentally used 3.6 volt battery for my mouse, now it only works with 3.6 volt
Without knowing the design, what really happened can only be guessed at.

I'm assuming that the mouse originally worked with only one 1.5 V AA cell, I mean, that it did not require two cells in series.

You probably damaged the DCDC boost converter chip inside the mouse. This boost converter is needed as 1.5 V usually isn't enough to power all circuits. This is an example of a typical boost converter circuit:



Note the inductor L1 and the diode D1, they are nearly always present in boost converters. It is possible that D1 is an internal (on chip) transistor with a diode in parallel.

If the IC is damaged and doesn't switch but L1 and D1 are intact then applying 3.6 V at the input will result in about 3 V at the output. That could be enough to power all circuits and make the mouse operate.