I am not a fan of Man United in the strictest meaning of the
term word, since I have been in Old Trafford only once.
Note the change to superlative there!
The use of meaning is correct here. The reference being to the meaning of the word fan.
I am not a fan of Man United in the strictest sense of the term, since I have been in Old Trafford only once.
The use of sense makes better sense here. The reference is to the popular idea of a fan, not the lexical definition.
Both words are correct. Depending on the context and the intended meaning, either word can find use.
Up to now we have considered the colors in the strictest meaning, namely as conditions or affections of the eye. This consideration is the first and most essential part of the color theory in the strictest sense, which, as such, must be the basis for all further investigations about colors and with which they always must remain in agreement.
Arthur Schopenhauer, Philipp Otto Runge: On Vision and Colors; Color Sphere (Google eBook) p.99
Note the use of both forms in the same paragraph, which may or may not be intentional, though.
In the strict(est) sense is idiomatic; in the strict(est) meaning is literal.