US leagues have salary caps built into their player union bargaining agreements and built into the league structure itself. In Europe where you have many teams, many leagues, and a wide variety of ownership groups (from millionaires up to literal country sovereign wealth funds), they maintain some semblance of "fairness" through financial fair play rules. This basically means that the teams are only allowed to spend on players (roughly) the same amount of money that they bring in as revenue. The revenue can be through tickets, concessions (I think), sponsorship deals, shirts/kits, etc... So in a very real way, having a big-money shirt sponsor directly impacts the quality of the players that a team can sign.

In the US, spending caps are set and enforced in other ways.