Thank you ronm for the great help!
As I found out now, it seems Xcode needs the Package being a Git repo in order to properly work, and the repo root directory has to be the root directory also for the package. (not in a sub-directory)
As for now, this is fine with me, as I plan to version-control the package anyway.
I first added the package via General -> Project -> Package Dependencies -> "+" Button
And then added it on the target as you suggested via General -> Target(s) -> Frameworks, Libraries and Embedded Content.
Topic:
Developer Tools & Services
SubTopic:
Xcode
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