How to start two independent incognito Chrome?

Incognito Mode Windows are limited to one per Chrome profile. To get more than one (without installing another instance of Chrome), you could add a new Chrome profile. More details here.

What you are looking for is SessionBox (link). It is the analogous of Multi-Account Containers for Firefox.

Each Box/Container stores cookies separately, so you can log into the same site with different accounts and online trackers can’t easily connect the browsing, i.e. for every browser window or tab you can log in to the same site with a different account.

You can customize tab's labels and colors to help keep your different activities or accounts separate.

Option 1: Use another, independant running instance of chrome, eg: http://portableapps.com/apps/internet/google_chrome_portable

Option 2: Setup another user profile and limit how long cookies etc are stored:

--temp-profile

On Ubuntu 18.04 Chromium 80, using temporary profiles instead of incognito windows worked:

nohup chromium-browser --temp-profile > /dev/null 2>&1
nohup chromium-browser --temp-profile > /dev/null 2>&1

If I login on Stack Overflow in the first instance, I still need to login into the second one.

As far as I can see, no data carries over from my main Chromium session into those, for example passwords are not remembered by the browser either.

Interestingly, also giving the --incognito option with --temp-profile appears to disable the --temp-profile, and logins are shared, which is unexpected.

See also: https://askubuntu.com/questions/304293/how-to-start-fresh-chromium-instance