The badger is a burrowing mammal of the weasel family. It’s also the nickname for a resident of Wisconsin.
As a verb, to badger means to harass or annoy someone persistently.
Fulminating about the canonical importance of Greedo not shooting first has metastasized into badgering fledgling stars off the internet for having the temerity to be female.
— Chris Baker, Wired, 18 Sept. 2018
While some animal names get used as verbs because the action is similar to something an animal does, that’s not the case with badger. More likely, it comes from the sport of baiting badgers, usually with dogs. (Appropriately, the verbs dog and hound can also mean “to harass or annoy.”)