This is most certainly the third installment.Prove me otherwise.
So by that definition, every form of media that uses the name of a previous product that was either the original or a sequel, also fits in the franchise.
The quickest comparison that comes to mind is the Halloween series of movies, in that series there was also a movie that did not fit the previous movie storylines at all, and yet was called a sequel because it had the same name and a sequel number.
I think it is in general disliked by all fans of that series.
I guess when it comes to considering Fallout 3 a sequel it comes to the opinions of the people who played the series, before Fallout 3 came out.
You can bring up the whole "Fallout 3" as much as you want but it is not a sequel to the series, rather than trying to connect the storyline even in the slightest way to the previous ones but expanding on it and making the game its own thing, it bluntly cut and copied and pasted elements from the previous games without any thought of trying to be original at all.
Super Mutants had to be brought back without any good connection to the previous game, other than them being a 'different strand' (which doesn't make sense as Mariposa was the ONLY source).
The Enclave had to be brought back despite that Fallout 2 makes it clear that they were defeated, gone, effectively eliminated as a grand power as the majority of their forces were eliminated along with their production capabilities.
The story was a mish mash of elements of Fallout 1 and 2 (get a GECK, save the Vault from thirst).
There was absolutely nothing good, original, or something that expanded on Fallout 1 and 2 while remaining its own thing.
I usually do not resort to this but as your posts follow a certain pattern I am going to ask; are you some Bethesda board member who has come here as part of an initiation rite, proving your undying love for Bethesda and their crappy game while sticking it to us NMA'ers which Bethesda wants to portray as backwards basement dwellers who fear and abhor any kind of changes?
Then get this; we don't mind changes or progress as long as it makes sense, what we do not like is being insulted, get the bird rubbed in our faces, or told what to like whatever the masses are pampered with at the moment.