The thing is, there's a sort of hierarchy towards bowl games. You have your legacy bowls that clearly are at a status above no other; the New Year's Six (Rose, Cotton, Sugar, Orange, Fiesta, and Peach) come to mind, and given that they are tied to the College Football Playoff, are also going to bring in the largest sponsorship deals for the bowl event groups, which is good for them (especially the Rose Bowl, which are undoubtedly the most traditionalist of the bunch, given how when the BCS was on FOX, the Rose Bowl maintained exclusivity on ABC due to having a separate broadcasting deal) but lousy for the company since they can't just rename the bowl game and get all of the advertising. At best, the company can get offhanded plugs by the networks or via camera footage, so it becomes more of a war of attrition.
It's when you start going down the list that companies can control the operations a bit more and even iconic bowls like the Gator, Alamo, and Independence aren't completely free from getting fiddled with. For a few years, the Gator Bowl was just the TaxSlayer Bowl; the Copper Bowl is now officially the Cheez-It Bowl, and the Citrus Bowl was for over a decade the Capital One Bowl.
Personally, it's at its funniest when you get to the absolute bottom of the bowl games; the events that play at the beginning of the cycle or have deals on CBS Sports Network. Naturally, the whole "DreamHouse New Mexico Bowl" fiasco was worth a few chuckles, but how about the St. Petersburg Bowl, where it was sponsored by Bitcoin one year? At this point, if a company decides to sponsor any of these crud bucket bowls, they have carte blanche to do whatever they want with it.