I know that it at least differs between the US and Europe. I've known ramp agents in a few countries and in the US the contracts in the end are handled by the airlines and training/regulation is governed by the FAA. The couple of European ramp agents are ultimately contracted to the airport (or really, the entity that operates the airport). This is a small sample, but there definitely exists more than one model of how ground staff is managed that is the norm. To further complicate things, the company that pays you may be a subsidiary of the airline you work for or sometimes a different airline or group of airlines, operating under a different name.

I'll give you an illustrative example that hopefully demonstrate how convoluted things can get. Swissport is one of the largest ground services providers in the world. It was first split off from Swissair as its in-house ground service department and became an entity under the holding company SAirGroup in the mid 90s, after plans to merge several of the flag carriers of smaller European countries fell apart when Swiss citizens voted not to join the EEA and therefore, denying Swissair of access to both 5th freedom rights in the EEA and potential cabotage rights. A few years later shortly after 9/11 the airline collapsed as did the holding group which resulted in Swissport being sold off to private equity (Swiss International took over operations on the airline side, sort of, by virtue of a Swiss government bailout and the acquisition of Crossair, the regional arm of Swissair that was divested earlier, by the creditor banks, but in a ton of debt and had to recoup as much as possible. Swissport continued operating after it was sold off and maintained relationships with, well, eventually just about every major airline through mergers and acquisitions, Crossair became Swiss International and was taken over by Lufthansa in 2005. Swissport has maintained relationships with Swiss through Lufthansa but only provides full service to Swiss at ZRH and cargo service at Basel (which Swiss pulled out of in 2015 on the commercial side, and also, is actually located in France, at least airside). It also handles regional operations for Lufthansa out of Munich. ZRH is the main Swiss hub and also a Lufthansa hub, but the fact that Swissport provides full ground service there is almost happenstance since Swiss serves over 100 other destinations and Swissport over 200, many of which overlap, but they do not directly serve the airline elsewhere except cargo operations to Basel. But there is another major airport in Switzerland, and Swissport operates there, but Swiss and Lufthansa Group uses dnata as their handling agent, a subsidiary of Emirates. dnata has no presence in Basel and is a minority operator in ZRH, where Swissport handles something like 80%+ of all ground operations. If you fly Swiss to just about anywhere else in the world, you'll see Swissport, but they aren't likely, save for some prior arrangement where airlines with limited presence may agree to contract through another airline, to essentially provide service in a separate agreement, but these are more ad hoc and subject to change and affects relatively few flights in comparison.

Lufthansa had its own ground services arm that it sold off - LSG Sky Chefs. Gategroup, which owns Gate Gourmet, was the Swissair catering service until being spun off in the same mid 90s expansion that ended up in the collapse. Swissair is no longer an active brand, but it is still valuable to an extent, so it is licensed out, although I have no idea where - like how Pan Am got licensed out to a freight rail carrier, it's likely stamped somewhere random and unrelated to Switzerland or air traffic, but brings in revenue so, why not?

I rarely fly now and when I do, it's domestic US and by charter. That's a whole other crazy mess but somehow, less messy than the scheduled airline industry and far less problematic than the duopoly both having supply chain issues, as both Airbus and Boeing currently does. That's on the news, if you haven't been paying attention, and it's not going away for a bit, by the looks of it. Air travel, meanwhile, remains safer than driving.