“Modern” Freemasonry (after 1717) has brought different sets of “Landmarks”, rules of the game so to say. One of these rules was that a Freemason had to be a man. This wasn’t such a weird rule in these days, since women didn’t even have the right to vote yet.
Another such rule is that a candidate had to believe in ‘something higher’. In some Grand Lodges this can be the Christian God, usually it isn’t that specific. The term “Grand Architect of the Universe” is vague on purpose. Both a Catholic and a Protestant can agree on the existence of a GAOTU, so can a Muslim, Hindu, etc.
But as society became more and more rationalistic, some people started to wonder if the requirement isn’t actually limiting? Can a person with no (set) faith not be a good man who can improve himself? Some think this is possible, so first in Belgium and later in France, the ‘religious’ reference to the GAOTU was skipped and more conservative Grand Lodges withdrew recognition. That means that members of these Grand Lodges can no longer visit each others formal meetings. Taken a step further, it means that some Freemasons will tell others that they are no Freemasons at all.
Regularity is a strange thing. Some Grand Lodges recognise only one Grand Lodge per country. Other Grand Lodges have no such limitation. Therefor it is possible that Italy has two “regular” Grand Lodges according to some “regular” Grand Lodges and only one to others. Regularity is in basis nothing but the mutual recognition between to organisations and doesn’t necessarily say anything about the recognition of GL#3.
The same goes within “Continental” Freemasonry. It is not so easy that all “Continental” organisation recognises every other.
Yep, Freemasonry is a weird and varied world.