Firefox

There are legitimate reasons. You may have a main account in a given page which you usually want to log into automatically, but at certain occasions you might want to use an alt without your browser forgetting to log into your main by default. A more specific example of this would be someone who has an alt account for NSFW stuff in social media because they want to be able to browse in their main without risk of genitals popping up on their screen while they're on the bus or using the phone next to family.

I use it to test that I've set up an authentication system correctly without a cookie bias, among other uses

Set everything up in main > confirm tests pass > log in from incognito with password vault to make sure the auto test didn't lie.

I use it when I need to log into a computer at work with a personal account, so that I know I'm logged out when I close the window.

anonymity is not the only usage for incognito, and frankly it doesn’t even make you very anonymous on its own

Signing in to a service would defeat incognito mode if its function were to make you incognito towards the websites you visit.

But since that's not what it does, I don't see why it would defeat anything.

Logging into alt accounts without logging out of main

My health and dental insurance are through the same company but on different accounts (they refuse to combine them), so I use a private tab to sign into both at the same time because it normally only allows one at a time.

Exactly because I want to log out of everything when I close my browser

You want to log into Facebook and as soon as you close it, it should be forgotten