Kansas City added a professional soccer team a couple years ago and they named it in a style more European. Instead of say “The Kansas City Kickers” they are called “Sporting Kansas City”.
This is probably a rabbit hole regarding the meaning of the term “official” but —
And from there— “© 2014 New York Giants” — if there’s any place that you want to be sure you have used your official name—or at least an official name, it’s in any legal or quasi legal usage, such as in a copyright notice.
The name of the corporate entity that operates the team is “New York Football Giants Inc.” but the name of the team itself is clearly “New York Giants.” That’s how it appears in nearly every format. It would be like saying that the official name of the New York Mets was “Sterling Doubleday Enterprises L.P.” It never was.
The team logos over the years have used various combinations of “NY,” “New York,” “Giants,” and “New York Giants,” never “New York Football Giants,” at least not in the modern era.
The heading at the top of the web page says “Official site of the New York Football Giants,” but it is then immediately contradicted by the title of the page itself, which is “Giants.com | The Official Website of the New York Giants.”
The heading at the top of the web page says “Official site of the New York Football Giants,” but it is then immediately contradicted by the title of the page itself, which is “Giants.com | The Official Website of the New York Giants.”
This one thing might make it ambiguous, but the weight of the evidence is that the name of the team is “New York Giants,” not “New York Football Giants.”
That’s too bad. They could have say an all-London team, maybe called the “Big Bens” or something, play say the Paris “Towers”.
That’s kind of funny. Imagine the teasm named after say a toilet manufacturer “The Flush”, or a toothpaste company “Cavity Fighters”.
One time Kansas City had a soccer team called “The Wiz” named after the wizard of oz. But they dropped it partly due to all the bathroom humor. Like the Wiz will be playing the Colorado Dump or the Portland Gas.
They then just became the Wizards.
All those teams still exist - though only London Scottish play in London. London Irish are based in Reading and London Welsh are based in Oxford. At lower levels, there are London clubs for all kinds of places. London New Zealand, London French, London Cornish, London Nigerian, London Polish, London Pumas (Argentina), London Wallabies (Australia) are all going concerns, at least last I looked they were. These are all actually based in London though the number of players they pull from the relevant communities varies (London Nigerian used to be mostly Nigerian and descendants of Nigerian immigrants - I reffed them last year and the side is much more mixed now).
I can’t imagine any common situation in which that would be confusing. There are hundreds of teams around the world that use names like Giants, Tigers, Lions, Bulldogs, Bears, Eagles, Reds, Blues, Warriors, Indians, Cardinals, Kings, etc. It’s rarely ambiguous in actual context which is being discussed.
Maybe they should be called the “AT&T Cowboys.”
The OP asked about teams in other countries that have the name of the city - they are named after the city, but they typically aren’t named after the mascot, are they? It’s not the Manchester United Jaguars or the Liverpool Titanics. Is the US the only place that names teams after a mascot?
That’s some full on RyanAir level of using “London” 
Then again Wasps just dropped the London and are moving to Coventry …
Some sports are, especially those that originate from North America. The local ice hockey team from where I grew up, for example, is the “Coventry Blaze”. I guess they could have gone with Godivas, but that could have got awkward. Not as awkward as Blitz though …
Going back to cricket, the team from where I grew up goes by three different names depending on what type of cricket they are playing.
First Class (4 days) - Warwickshire
50 Overs (1 day) - Warwickshire Bears
T20 (1 day, shorter game) - Birmingham Bears
Why Bears? Well the flag and coat of arms has a bear on it.
Oh man, we totally should rename Coventry City to the Coventry Elephants.
In cricket, the English county teams also play in a limited overs competition, which is a shortened version of the game that was designed to appeal to younger audiences and bring in more supporters. In 2002, the traditional county teams incorporated “nicknames”, presumably again to appeal to the youngsters.
Some of these nicknames follow an American mascot system - the Sussex Sharks, Leicestershire Foxes and Glamorgan Dragons spring to mind. Others - such as the Worcestershire Royals, perhaps - didn’t.
Whilst there are many football (soccer) teams that have a city name in their title, it doesn’t necessarily mean they are the only or main sports representative of that city - Manchester has two top flight teams who compete for that honour - Manchester United and Manchester City, and of Birmingham’s two leading teams, the biggest is the one without Birmingham in its name (they are Aston Villa and Birmingham City. Aston is an area in Birmingham).
Shell (the petroleum company) fielded a team that for a while was called the “Bugbusters” as they were trying to market insecticide. There was also the Sta Lucia Realtors. For a team name, though, nothing beats the Purefoods Tender Juicy Hotdogs.
Well, in the USA, no one city has more than two teams in any major athletic league.
New York City and Chicago have two major league baseball teams. Los Angeles has two NBA basketball teams. Northern New Jersey has two NFL football teams. But otherwise, most big cities have only one team in each major sport.
But how many pro soccer teams play in London and/or the suburbs of London? Quite a few. So, while it’s fine for an American baseball team to call itself “The Boston Red Sox,” and to position itself as Boston’s baseball team, it would be difficult for Arsenal or Tottenham Hotspur to market itself as THE London team. If anything, it would seem rather arrogant to fans of Chelsea or West Ham.
Worcestershire Royals fits the pattern just fine. We have a Kansas City Royals and in the past had Cincinnati Royals.
The pattern is [geographic indicator] [nickname]. The nickname can be a mascot-type animal but doesn’t have to be.
You could argue that New York has three NHL teams and, before the Whalers left, four.
But you have limited your scope to teams in the same league. In England, they’re counting the teams at all levels, not just those in the top league.
To make an even comparison, you should include, taking baseball as an example, the New York Yankees, New York Mets, Brooklyn Cyclones, Staten Island Yankees, and the Long Island Ducks.
Los Angeles Dodgers/Angels, New York Yankees/Mets/Giants, Chicago Cubs/White Sox, St. Louis Cardinals/Browns, Philadelphia Athletics/Phillies, Boston Braves/Red Sox. And going back to the 19th century, when these naming conventions were adopted, places like Baltimore, Philadelphia, Washington, New York, and Chicago could have three or more major clubs competing professionally.
None of these clubs felt a hesitation to claim to represent an entire city, even when there were multiple clubs in the area.
So it’s not about the structure of the system or the number of teams.
And pride and arrogance and puffery are an integral part of sporting. Why should the hypothetical London Chelseas care if the fans of the London Arsenals think they’re arrogant for claiming to represent the whole city?
And often, teams are known for shortenings or nicks which don’t mention the city, whereas the full name does. For example, Real Madrid (known as Madrid) and Club Atlético de Madrid (known as Atlético, not to be confused which Athletic, which is from Bilbao and whose official full name is Athletic Club) are both from Madrid, as is Rayo Vallecano aka el Rayo (Vallecas is an area of Madrid). Also, often a team is so closely associated with a city that many people don’t realize the town’s name isn’t part of the club’s name.
Argentine football has some weird names. River Plate. Racing Club. Newells Old Boys.
New York City now has two basketball teams as well.
No, that’s the nickname.
There’s also the Minnesota Twins; teams in Minneapolis use the name of the state since there’s a rivalry with the city of St. Psul across the river, When the Twins moved, no one from St. Paul would think of rooting for a team named for Minneapolis (the minor leagu Minneapolis Millers and St. Paul Saints had a longstanding rivalry at that point).
I was specifically answering a question about mascots, but all the County teams fit that pattern of <county><nickname>.