@Till Blesik I'm glad that you were able to resolve your issue and thank you for posting your solution so that others experiencing the same thing can easily reference this! Since the Microsoft Q&A community has a policy that "The question author cannot accept their own answer. They can only accept answers by others ", I'll repost your solution in case you'd like to "Accept " the answer.
Issue: Encountering difficulties while attempting to upload files for creating a vector index in cognitive services. Despite efforts to create a data asset or run the vector index flow, the upload process is not functioning. Clicking the "+" button or attempting to upload directly to a blob store does not yield any results. Additionally, using the storage explorer to upload files also proves unsuccessful, as the link to open a local file selection does not work.
Solution: After checking the browser console, it was apparent that certain parent elements were missing to create the file picker. However, resolving the issue involved updating the Edge browser. This suggests that a recent update in the Azure Portal might be incompatible with older Edge versions. Notably, multiple team members experienced the same problem simultaneously, and updating the browser resolved the issue for everyone.
Please remember to "Accept Answer" if any answer/reply helped, so that others in the community facing similar issues can easily find the solution.