Have you played with the app directly? If you don't have a sub, you can't get to the data sources to see how it all fits together and can be individually corrected. It's $10 for a year, and there's a 30-day free trial, so if interested - you can see if it's useful for a few weeks before cancelling without any cost. But here are the app screens that point to team data. Let's take this example team:
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That team has a couple of different events/tournaments. Scrolling down, here are all of their games:
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If you click over to Sources, here's the actual data sources that they were coming from:
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You can see how the names aren't the same. The first two were probably automatically matched, while the 3rd was potentially linked together manually by someone, as they had used a slightly different name for that 1 tournament.
Here's that specific tournament:
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If it turns out that I was familiar with the team - and knew that it actually wasn't them - I could delete that data source by hitting the trash bin next to the invalid source:
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and then hit Done in the top right, and the team data would then be corrected. That's really it. You generally can't enter your own data in - the only option is manipulating the data that SR has pulled in via these game sources, and adding/deleting them to the team entity as appropriate. If I knew there was another tournament out there with the team named somewhat different, I could hit Add Sources, search for something close to that other team name, and if it was the right data I could add it to this team entity and then save it the same way.
The workaround being proposed for MLS is interesting, and I do wonder about how the UI will accommodate creating some new data entries, and how that will be monitored over time. But I have no doubt that in this same manner - if there is bad or even malicious data, it will be able to be removed/changed by the next person that comes across it.
Now your specific example does have its conundrums. If a team really is named "The Super Bunny Puppies Academy Premier Elite", and they use that exact same name in everything they enter - but in some games it's made up the team A starting powerhouses, but in some games it's entirely made of the scrubs that are most worried about splinters from the bench - it could be hard to separate them. The first fix is to have the club registrar and/or the team manager realize that using the exact same name for essentially different teams is a bad idea for a host of reasons, two of them having a consistent game history and having objective measures about how good the team actually is - all helpful when deciding what brackets to enter for each event, and even how to place the team next season. If the team does start naming more consistently interpretable as they enter events/tournaments, and now there are "The Super Bunny Puppies Academy Premier Elite Red" and "The Super Bunny Puppies Academy Premier Elite Blue", once their first data starts showing up in SR - you can then pull in the correct events from the prior combined name into the appropriate new team entity. For most individual tournament info, this would work fine. Though it may be tedious to figure out which is which, it's doable, and it's the very foreseeable result of using the same team name to represent multiple things at once. The sticking point that isn't resolvable, is if all of some particular data is in a single source, and that single source is actually representing multiple versions of the team, with the same name, in the same source. For example if this was in league - which typically only has a single data source entry for all games for the season (but still often split by fall/winter/spring/etc.), there's no way to say let's apply games 1,3,5 in league to the Red team and games 2,4,6 to the Blue team.
The good news, is that if a team has been sharing an identical team name across non identical teams, and at some point decides that they don't want to do that anymore - starting with a new more intelligible team name for GS/SR whatever, will still result in a rating showing up for the new team name in only 6 or 7 games. One of our teams had to do this just this year, when there were some name swaps in the club from X to Y. The Y team had no rating - but within 4 weeks - they not only had a rating - but it was identical within a tenth of a point to their rating from all of the prior game data in X. It's a surprisingly small amount of rated games that can be relied upon for the rating to show up.
If you do have a specific example team that you're aware of, and wondering if it is possible to clean it up or not - PM me and if you want I'll send you screenshots of exactly what the data looks like as is, and potentially recommend how I'd fix it if I knew more about the team itself.