User lacks permission to complete this action. You need to have 'AddPackage'

It seems I fixed by adding to the feed's permission settings the Build Service as Contributor.

It's a bit confusing tbh but it's now working fine.

For those who are completely lost like me, the page that diegosasw said is in the Artifacts menu, not in the Task or in the Pipeline:

Great find! Allowing project-scoped builds in the ... menu worked for me.

But in order to be able to push the package from Azure Pipelines to the Azure Artifacts feed at all, I had to add the same feed as Target feed under the dotnet restore step before dotnet build and finally dotnet push. https://docs.microsoft.com/en-us/azure/devops/artifacts/nuget/dotnet-exe?view=azure-devops