That's what warakorn was saying, that while EU261 doesn't apply the OP could attempt to enforce a larger financial compensation under German common/contract law in terms of the conditions of carriage. I don't think the destination of the flight in question being HAM is relevant, since the decision to downgrade and acceptance thereof (by checking in and boarding the flight, even if not aware of the downgrade at the time) took place outside Germany. However the purchase presumably took place in Germany and one could argue therefore the contract falls under German jurisdiction. I think this is a long shot and not worth the hassle though.

Let's look at the numbers, typically on flights to CPT in my experience if we take Y=1 (a typical flight in Y as a base) then J = 3Y and F = 2J or 6Y. This could go up or down of course.

The great circle distance of HAM-DXB-CPT return is approx. 15,544nm, of which the DXB-HAM leg which we are considering is 3037nm, i.e. 19% of the total distance.

If we accept that F = 2J then OP is entitled to a refund of 1J, or 19%/2 = 9.5%. So to be completely honest, Emirates' number isn't completely unreasonable. Given fluctuations in pricing it is conceivable that 8% is in fact the correct number. Of course the correct figure could also be 12%.

All of this also does not account for the inconvenience and disappointment which I think EK should address in a separate compensation in the form of Skywards miles, especially if OP did not receive any communication. If booked via a TA however then Emirates shouldn't be liable for any further goodwill compensation as the onus is on the TA to inform OP and should have known that rebooking on another flight is possible.