the en-gb language is not available on its own as an installable pack - it is simply part of the core installation.

I have never heard of this error message that you are getting and have not seen any reports of this either. Can you post some screenshots of the error

You're welcome, Brian, I've tried so many things already.
It can only be an error in the database that I just can't find or have no idea where to tackle the problem. Has no one else had this problem of not being able to install languages ​​in the backend? I just can't imagine that. What's much worse, I think, is that a problem like this is carried over from J3 through 2 migrations to J4 and then J5 and possible errors are then not eliminated by the updated core. Everything else, like installing from the web catalog or installing extensions, works without any problems.

I can also provide a backup of the J5.1.3 installation if you want to investigate it. I can also provide a backup of the old J3 installation where this error already exists.

I found something here but it didn't help me either.
https://www.web-consultant.at/joomla-cms/joomla-anleitung/problem-beim-sprachen-installieren

The information in that post makes sense and its cause (a faulty template custom install) makes sense to me. Are you sure you followed all the instructions in that post? If you send me a backup I can take a look when I get the time

Yes, I've tried everything but have never experienced anything like this before. I'll send you a link to your email address to download the backup including the DB.

I found the problem

Your list of update sites should look like

You do not have an entry for the en-GB language pack

Go to your database and the table #__update_site_extensions

You should have TWO entries that have the same update_site_id but I am guessing you only have ONE for the GERMAN Language but nothing for the english

Go to the #__extensions table and find the entry for the english language pack and take a note of the ID. In my case it is ID 240

Go to the #__update_sites table and find the id for the accredited joomla translations. In my case it is ID 9

Now go back to the #__update_site_extensions table

If you have a record with the extension id for english then it incorrectly will not have the correct update_site_id (in my case the correct data is 9 and 240. If that is the case then edit the database record

If you do not have a record for the english language then create a NEW record in that table with the ID values you found earlier

this should be closed - it is not a core issue.

The plugin pkg_en-GB is not present in the database.
This will be the cause.

Thank you very much Brian. The problem is now solved. I looked at the missing row in the table in another database and then inserted it 1:1 as a new row in the affected database (table ..._extensions). Then updated the update sources and it worked without any problems. So someone simply deleted the en-GB plugin in the old Joomla 3 version. You never stop learning.