Try this:

version: '3.7'

services:
    api:
        image: 'gthole/drink-stash:latest'
        init: true
        restart: 'always'
        environment:
          - SECRET_KEY=YOUR_SECRET_KEY
          - ALLOWED_HOSTS=SYNOLOGY_HOSTNAME_OR_IP
        ports:
          - 8081:8000
        volumes:
            - drink-stash-data:/data
            - drink-stash-public:/public
volumes:
  drink-stash-data:
  drink-stash-public:

This will create volumes instead of mounting the folders. Mounting the folders (which is what you tried doing before) should be possible and nicer to use since you would be able to navigate the files directly, but since I do not know the filesystem and layout of synology it’s harder for me to help there. Using named volumes like I just sent you should work for any filesystem/layout so you shouldnt have any problem with that.

Also, use the synology hostname and/or ip for the allowed_hosts, localhost would only work if you were running that on your computer. The app should then be available at http://SYNOLOGY_HOSTNAME_OR_IP:8081