What else I tried: another version of the engine, other plugins, clearing the cache, re-logging into the account, checking for installation in “Installed Plugins”, checking in the Unreal Editor.
Ok, I fixed it myself. The order of actions for reproduction: On account 1 download the engine, go to account 2 - you will not be able to install the plugin on the engine. At the same time, if you install not a plugin, but, for example, an asset that creates a separate project, when you try to install it will give (finally) an error that the engine is not installed. You need to verify one of the engine versions, then all versions will be detected and the plugin will be installed successfully.
I have the same issue, tried verifying engine files, it did the job, but I still cannot install any plugin.
When I try to add anything to the project it seems that the launcher still doesn’t recognise Unreal Engine being installed.
Though I do can launch the engine itself with no problems
Before verifying the engine, I also cleared the launcher cache along the path: C:\Users%username%\AppData\Local\EpicGamesLauncher\Saved\webcache_4430
And one more thing, when I verified the engine, its files were slightly modified, so the launcher had to download about 400 MB, maybe this also affected it. Alternatively, you can delete a couple of files from the engine folder so that the launcher has to download something for the engine.