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@john_2141 

As has already been said - you can do OCR in multiple ways - but you'll still need to position texts manually anyway...

You could probably make your life a bit easier - if those vectors has a unique formatting - always the same for all countries - then you could find them and move them on a separate layer - in Illustrator - then hide everything else, export only this one layer and run through OCR.

The whole process could probably be automated ... each vectorised name exported as a separate image - run through OCR - maybe somehow original name could be paired with recognised text then correct text placed back in Illustrator - or InDesign - but you rather don't have time for that?

The problem is that the names of the countries are only available as vector files, but not as text. But I am supposed to hand in the whole file as an .indd file with the names of the countries as text. Does anyone know of a tool that can automatically detect text...

By @john_2141

There are multiple tools that can detect text on an image. For example, if you're on Mac, you can save your map as an image, then open it Preview, then select and copy your country names as text. This at least will save you the time of retyping them manually. You'll still have to do everything else manually of course, such as positioning them in Illustrator/InDesign, applying fonts etc.

It's quite possible that more robust AI-based (not meaning Illustrator) text recognition tools also exist that can save you a few more manula steps.

Illustrator ist an und für sich KEIN allgemeiner PDF-Editor, doch hier würde ich es auch damit tun, weil die Texte bereits als Outlines vorhanden sind.

Texterkennung auf einer Landkarte wird kaum funktionieren. InDesign kommt da nicht infrage. Mach es in Illustrator, auf einer neuen Ebene und gib acht, dass für zukünftige Überarbeitungen der Text auch Text bleibt, natürlich auf neuen Ebenen.