According to the Online Etymological Dictionary, matinee was a mid-19th century borrowing from French, and the word carried the sense "daytime" rather than "morning"--see here--so the question then becomes, why was the word at that time no longer limited to the meaning "morning" in French?In English we use Matinée show for a afternoon show (around 3 PM.)
But Matinée (comes from French : Matin : morning ); so how come it reffer to an Afternoon show?
Well, not according to the source I cited, in which, in the sense borrowed, the reference was already to daytime rather than morning.It arrived still meaning a morning entertainment, but over time was extended to anything other than evening.
And of the matinée musicale:An entertainment presented in the morning or (in later use) at any time of the day before the evening; (now) esp. an afternoon performance at a theatre, cinema, etc
A musical entertainment given in the morning.
Absolutely correct when the origin of French matinée is concerned.Webster's Third has the etymology as "French matinée morning, time of day before dinner, matinee, from Old French matinee morning...." I wondered what dinner referred to there, whether to the midday meal (as it was in rural Illinois when I was a boy) or to the evening meal. I got the answer in The Century Dictionary, in which the following note is made in the entry for matinee: "The general dinner-hour of early times having been at the close of the forenoon, the French matinée, like the English morning, is often considered as extending to the common modern dinner-hour in the evening, especially in cities."
So, if I am interpreting the entries from the three sources correctly, before the word was borrowed into English from French, matinée already meant "the time before dinner," and since, in French cities, dinner had already come to be the evening meal, matinée had, by way of matinée musicale and even before it was borrowed into English, gained the meaning "a theater entertainment occurring before the evening meal."
Adams' New Musical DictionaryMatinee : A musical entertainment given un the early part of the day ; a morning concert
Matinee musicale : a morning concert