Due compatibility table we can use
JSS 16 only on
I have no ability to upgrade Sitecore, so my question - Did someone tried to install
JSS 16 on Sitecore 9.3, and is it possible or totally impossible
Welcome to Sitecore Stack Exchange. I wouldn't recommend trying to install unsupported JSS version, even if you succeed on making it work if you find any issues related to the module you won't be able to get official help for example from sitecore support.
Depending on your current website installation (mostly if you need to migrate xDb data) I would rather explore the possibility of upgrading to sitecore 10.1 as the process has been simplified since the older sitecore versions. You can take a look what steps would be required in the upgrade guide https://dev.sitecore.net/Downloads/Sitecore_Experience_Platform/101/Sitecore_Experience_Platform_101.aspx section 4.2.6.
I noticed on your tag
Next.JS I guess you want to use JSS with NextJS.
I am pretty sure next.js will not be fully compatible with
My recommendation is to
upgrade your solution to 10.1.
Sitecore support will not help you if some issues appear because you are using not recommended versions of a module.