Apparently that's what some folks on the left think. Folks on the right rush to defend a product which may very well be toxic.
This is just another example of the dog and pony show which is modern politics. You are presented with a choice: either be a feminist and blame everything on the mythical "patriarchy" or be a consumerist and support woke corporations. Of course, it's a false choice, because you don't have to be either of these things, that is, you can freely reject both.
Here I'm going to make a confession. I don't use a deodorant. Haven't used it for like 10 years or longer. I don't stink. I shower and change my clothes regularly. Shaving your arm pits, especially in summer, helps, too. BTW, I recently switched to showering with warm water only (no shower gel or soap, just like cavemen regimen suggests) and I haven't noticed any difference. Warm enough water and some scrubbing do wonders, unless you work in a coal mine.
So no, deodorant isn't a tool of patriarchy. The US of America isn't patriarchy by any long stretch. In fact, it's like the ground zero of feminism. But, it's also the ground zero of consumerism. And corporations don't really care about your well-being, they only care about making money. I'm quite proud that I'm cheating them out of some of it:)