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That's not right. Chrome's incognito mode is per window.

Not even. Chome's session isolation is only per each of the two browsing modes.

Chrome supports multiple profiles (click the person icon or username on the upper right), not just logged and incognito.

Yes, but all incognito tabs are in the same session, even if opened in separate windows.

I don't think those profiles let you do things per-tab.

This. The relevant Chromium issue report is here [1], and it's been open for years.