I’d call it aggressive moreso than invasive. When I think invasive I think of something that can propagate and spread multiple ways. Running bamboo has aggressive rhizomes. As long as you properly plant it with proper containment methods (bamboo barrier, etc) it can be properly contained. Then you have the clumping bamboos that don’t spread much at all. My bamboo is in my chicken run as I’m trying to give them some form of grass that is representative of their paleolithic history. They all originated from the tropical red jungle fowl. So yeah, you can’t just throw it in the ground and let it go out of control. And up here in zone 5 it takes quite a beating.
Also, there is plenty of bamboo that is native to the southern US. That rivercane I just received is all US native and naturally grows near bodies of fresh water in the SE. It is actually important to the wildlife and ecosystem down there since it remains evergreen all winter. Unfortunately people have been wiping it out over the past century.
But even out of control bamboo can be controlled. If you keep mowing down the soft new shoots each spring eventually the rhizomes run out of energy to keep putting out new growth. It takes a few years, but eventually it gives up.