And I have found several uses of the verb "jogar" applied to the practice of the montante, some quick translations " MONTANTE Very large sword that is thrown or "played" with both hands", "Montante is a sword, that exceeds in its size the height of a man and is "played" with the two hands"
Diccionario da lingua portugueza, Volume 2
Officina de S. T. Ferreira, 1789
"MONTÁNTE, s.m. Espada mui grande, que se mandava, ou jogava com ambas as mãos, e por alto. Miguel de Arnide era tão agigantado, que trazia na cinta um montante por espada ordinária."
Revista de Portugal:
Língua portuguesa, Volumes 7-8
Tipografia de Editorial Império, Lda., 1945
“Montante é espada, que excede na sua grandeza a altura do homem, e se joga com duas mãos”
Museu literario: util e divertido, Issues 1-13
"Outra observação devemos fazer no jogo desta espada: imaginaremos nós que he huma espada commum? Pois enganamo-nos nós que não he senão hum montante, he huma espada que se joga com duas mãos"(...)
I don't mean that this proves or even reinforces much, a connection of jogo do pau with the montante, however, I think it reinforces the idea that the expression used for staff fighting in Portugal, and in other places in Europe(spain and france at least), derives from what was understood to be a proper and serious way to use a weapon/sword, and not a game or a modification of fighting for ludic purposes, as a dance, for example, can be, if derived from a form of fighting as are some folk dances.
I have seen people quoting with the expression playing, referring to sword practice, with wood swords, form italian and german books, don't have it now but I can look for it. It seems to be a quite spread use of the expression referring to martial practice. Also, Japanese martial arts seem to use it too. "shiai"?