Yes, you can have a push notification system set up for your Feed.
The Stream Feeds API, does support a solution to push notifications, it is just not "built-in" as it is for the chat API.
It will require you to set up your own push provider (Firebase for example) and connect to Stream via SQS or Webhook.
Stream's Firehose allows you to subscribe to all feed updates and have them shipped to your own infrastructure via different transports. These transports make it easy to extend Stream.
Realtime transports are useful for:
push notifications using your own infrastructure
send emails based on user behavior
do offline processing