[ Wiktionary for 'même' :] Etymology
[3.] From Middle French mesme, from Old French mesme, earlier meïsme,
[2.] from Vulgar Latin *metipsimus [= the same],
[1.] from Latin -met [emphatic suffix] + ipse [“himself”] + -issimus [superlative suffix].
Cognates include Spanish mismo, Portuguese mesmo and Italian medesimo.
I added the Syntatic Categories for 1 and the English meaning in 2, from the link in 2 above.
How did 2 semantically shift to 3?