According to Merriam-Webster, the relevant definition of "franchise" here is "a series of related works (such as novels or films) each of which includes the same characters or different characters that are understood to exist and interact in the same fictional universe with characters from the other works." Wikipedia similarly defines it as "is a collection of related media in which several derivative works have been produced from an original creative work of fiction, such as a film, a work of literature, a television program or a video game."
In other words, in the context of video games, it's an actual series of games. So, I will exercise my right to be pedantic and exclude one-off games. I will also exclude actual game series that are known mostly or entirely from just one single entry, with its spin-offs having been largely ignored or forgotten.
If I had to make a Top 10 all-time list of major game franchises based on a combination of popularity and impact on the industry, it would be:
1) Mario
2) Pokemon
3) Grand Theft Auto
4) Call of Duty
5) Final Fantasy
6) Halo
7) Street Fighter and Mortal Kombat (tie)
9) Sonic the Hedgehog
10) Resident Evil
Mario & Pokemon being the top two is obvious. Mario's impact and popularity can never be overstated. Same for Pokemon. But Mario gets the #1 spot for making Nintendo a household name. Mario has been in more games than perhaps any other character, games with his name in the title have sold more than any other game series in history, and both Super Mario and Mario Kart games are consistently among the best-selling games every generation. Without Mario, it's entirely possible that consoles may have never regained popularity after the Crash of '83.
GTA has produced multiple mega-blockbusters, and it paved the way for the dominance of open-world games in the single-player space. Just five mainline entries in the series (III, Vice City, San Andreas, IV, and V) have a combined 270M copies sold. GTAV by itself represents at least 180M of those copies, making it the second best-selling game of all time after Minecraft, which means its therefore the best-selling full-price game.
CoD warrants a place in the Top 5 simply because of its sheer ubiquity, the annualized series being by far the most popular FPS franchise. Each game in series has since as least as far back as 2009 been the best-selling game in the U.S. every year that didn't have a major Rockstar title released, plus some of its gameplay trappings became somewhat commonplace in FPS titles, namely perks & killstreaks.
Final Fantasy is not just the most popular in its genre, but it also made JRPG games a global phenomenon, and it set a lot of standards for its genre. The original game saved Square from going out of business, and FF7, the first mega-blockbuster in the series, did more than anything else to put PlayStation on the map, drawing significant mainstream attention to Sony's fledgling console brand, with sales of the console exploding after the game's release.
Halo not only was single-handedly responsible for making Xbox a relevant brand, but it also created or popularized many gameplay elements (e.g., dual-stick controls, splitting melees & grenades from weapons) that are still the norm in the FPS genre, and broke new ground for level design in a genre where small, claustrophobic, often maze-like levels were the norm.
Street Fighter II and Mortal Kombat were released only 18 months apart from each other, and were both massively popular and massively influential, dominating arcades in the early 90s and spawning massive media franchises. While one-on-one fighting games existed well before either one of those games (the 80s had Karate Champ, the original 1987 Street Fighter, and even boxing games if you want to include those), they were the ones that popularized the genre, to the point where countless other series and individual titles popped up in their wake throughout the 90s. Both series established many fighting game norms, and Mortal Kombat's bloody violence, along with that of Doom, helped spawn the ESRB.
Sonic did for the Sega Genesis what Super Mario Bros. did for the NES six years earlier, or what FF7 did for PlayStation and Halo did for Xbox. Like those other games, it was the killer app for its platform brand, and Sega owes their brief but major success in the console market to the blue hedgehog. Despite Sega going third-party, Sonic remains popular to this day, and Sega is still identified primarily with the series. The series' success also resulted in a plethora of mascot-themed platform games throughout the remainder of the 90s, some of which still retain relevance to this day.
Resident Evil is by far the most popular survival horror series (and most popular horror series in general), and it was the series that popularized the genre. It is Capcom's biggest franchise by far, and the third most popular non-Nintendo Japanese video game IP after Final Fantasy and Sonic.