The term "licentiare" comes from medieval, not classical Latin. The OLD, as fine as it is, is mainly useless for medieval latin vocabulary. Better try the Dictionary of Medieval Latin Dictionary from British Sources (DMLBS), freely available at Logeion (https://logeion.uchicago.edu).
For "licentiare" (you must search for verbs in the infinitive, not in the 1st person present, as in most other Latin dictionaries), the DMLBS shows entries as early as 12th century. See here: https://logeion.uchicago.edu/licentiare
The DMLBS' third sense of the term is this:
3 to give leave (to depart), dismiss. b to send on leave, grant
furlough, to disband (army); (~are ad propria) to send home. c to
dismiss, expel (from office, service, community, or sim.). d to remove
or expel (from one’s land), dispossess. e to dissolve (parliament);
cf. departire 2a.
which seems akin to the French "licencier".