Thanks for your comments.

I finally got it working by using
az functionapp config appsettings --settings "@appSettings.json"

One mistake I made was that my filename was appSettings.json and I wrote AppSettings.json in the command. The error message I got then was "Failed to parse string as JSON: /azp/_work/2/s/deployment/workflowparameters/Dev/AppSettings.json" so I thought it did not understand that I wanted to use the contents of the file. It seems like you get that error message also when it can't find the file.