"I could be something approaching myself''

"Approaching" can literally mean "nearing", and in this kind of context it means "almost" or "only just".

For example, this quote:

"It had been a while since I had been anywhere with food that I deemed approaching edible."

Here the author is describing food as "approaching edible" to mean it is only just worthy of eating.

So, in your example:

Partly this was because she was the only person on the island with whom I could be something approaching myself.

This means that the first person has not been able to 'act like themself' with most people, but she has found someone with whom she can almost be like herself, ie she still has some reserve perhaps.