Can I fly from Belgium to Germany with Belgian Residence Card Type E for identification?

Travel documents for EU nationals

If you are an EU national , you do not need to show your national ID card or passport when you are travelling from one border-free Schengen EU country to another.

Even if you don't need a passport for border checks within the Schengen area , it is still always highly recommended to take a passport or ID card with you, so you can prove your identity if needed (if stopped by police, boarding a plane, etc.). Schengen EU countries have the possibility of adopting national rules obliging you to hold or carry papers and documents when you are present on their territory.

At the Brussels airport, you will never need to show your ID when traveling to Germany. You just swipe your boarding pass (print it ahead of time) and go through the automated gate. They will not check at the gate either.

Same goes for Hamburg. I have personally flown the route you are describing without ID, with two caveats:

1. If you check luggage they may ask to see your ID.
2. As the above answer states, technically you need valid ID in Schengen because you can be stopped at any time. That has nothing to do with the aviation industry.