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Only in Chrome? In every browser using private mode, private mode only delete the local storage (wbSQL, Serviceworkers, cookies, cache, etc), no other things, it hide nothing, for webpages which log you (or the search engine you use, AI and some other extensions which you use in "private"mode) it's irrelevant if you use private or normal mode. It's a very frecuent missconcept to believe that the private mode is the same as anonym browsing, simple extensions, like Cookie Autodelete or SiteBleacher do exactly the same as browsing in private mode, but with the feature that you can partial or full whitelist the pages where private mode isn't needed.
More or less Private only if you use VPN, SPN, MPR, Snowflake or at least a proxie.
I only mentioned chrome due to the recent shenanigans with their "incognito mode".
Well, all browser have incognito or private mode, it's nothing special. Vivaldi in this moment has released in the last snapshot an inbuild MPR in test, this will be a real private incognito mode.