Most certainly not.
This article on Jogo do Pau specifically tells a story of how stick fighting naturally evolved in rural areas as it is both a useful thing to have in general in the wilderness as well as a weapon simple enough to be available for everyone while not being perceived as an expression of possible aggression. And this extends back into the Neolithic Age:
Human beings have always had to fight to survive and humans have always employed tools. The simple stick was almost certainly among the first tools to be turned to martial purposes, as an instrument of attack and defence against animals. As societies evolved from the nomadic hunting and food gathering stage, conflicts arose; competition over resources, etc. boiled over into personal combat, and people created series of specific movements, attacks and defences, with their utilitarian sticks.
As of the very peculiar style, this is bound to the properties of the wood used as is pointed out as well:
In Portugal one very rich technique was developed, adapted to a type of wood known as o varapau or cajado. As with the development of staff weapons in other countries, the pau was also part of the normal equipment of the field-worker, used as a walking stick or hiking staff and as an elementary weapon of self-defence itself against the aggression of people and animals.
This very good, long, and fully sourced article on Irish stick fighting is similar insofar as it involves "stick fencing" as a translation of the original term. But it also points out how this is misleading because of the different techniques and esp. handling of the weapon (balance, range, and areas of defence being different):
Boiscín can be translated as “fencing” according to the 1977 dictionary of Niall Ó Dónaill. This relation to fencing does not necessarily means a connection to sword-fighting; as fencing could be used at the time to denote other martial arts aside from swordsmanship.
and, more to the point:
We know that French La Canne was inspired by saber fencing, and both are very similar, and so if Irish stick came from the same source it would probably also be similar, but then we have absolutely no source that mentions an Irish style which would resemble saber fencing.
On the general history of stick fighting in Europe:
What is much more interesting is to observe that shillelaghs like clubs were once extremely popular all over Europe, at least until the 18th century. The imagery of similar clubs is found in Ancient Roman mosaics, medieval illuminated manuscripts and even the art of William Hogarth. But by the 19th century, only the Irish and Britons still used them, the later abandoning them rather early. Would this be indicative that the shillelagh would, in fact, be the remnant of an ancient type of vernacular martial art which survived on the Island as other European traditions such as the bagpipe did?
TL;DR
While there are forms of stick fighting coming from the "nobler" sword fighting like the French La Canne, it is much more probable that stick fighting in general developed in all cultures before the Iron Age as means of self-defence both against humans and animals. The very special style of Jogo do Pau is due to the combination of length and flexibility of the wood used.
