A nice point, Dr J, but perhaps over-classical!
I should perhaps have mentioned in my opening response above, Uzoma, that Chambers comments re 'mentee' that it is (quote) "non-standard". It seems, therefore, that even its editors were not wholly convinced that it was a fully-accepted word in British usage in 2003.
It first appeared in America in the mid-1960s, whilst the latest print edition of The Oxford English Dictionary came out in the late 1980s. Clearly - given that that dictionary does not give it house-room - it had failed to catch on here in a quarter of a century. I myself have not heard it used to this day in Britain either, although it now is listed in the online OED.
As far as I am concerned, we all have as much right to use 'mentee' to refer to someone sent to a mental institution or even someone mentioned!