Hi Patrizia Hörmann,
Thank you for the posting in our forum community. I'm truly appreciating your cooperation in our forum community.
Regarding your description, I’d like to share with you such kind of information that “asana” is third party app in Microsoft Teams. So, according to your above mentioned screenshot, at once kindly try to double confirm that if the Third party apps option is allowed under Third party apps option from Microsoft Teams Admin Center or not. Because if the third party apps are not allowed in Microsoft Teams, then it might be the possible cause that we cannot add this app in tab in Microsoft Teams.
Here are the steps and screenshot for allowing Third party apps for Microsoft Teams:
Go to Microsoft Teams admin Center > Select Permission policies under Team apps > Click on Global Org-wide default) option > click on Edit option > Select Allow all apps under all options >click on Save.
Note: if you have not permission to access of Office 365 Admin Center, for performing this above steps activity kindly contact to your Office 365 Admin of the organization.
Note: After performing above activity, kindly wait for few hours and after that try to check the result by searching apps in Tab in Microsoft Teams.
Hopefully above information will work for you. Kindly feel free to contact us with your updated result.
Regards,
Darpan