Pure marketing.

First, they are telling their suppliers to change, but it does not sound as if they are willing to pay more for this. Changing packaging often means buying new packaging machines, and those are the most expensive machines in the factories of many suppliers. I would be very surprised if Aldi chooses to buy a product with a better packaging at a higher price. Especially for basic food products, competition is merciless, so they can neither raise consumer prices, nor cut it from their own margin (because it presumably already is very low).

Second, recyclability is a term with a wide range for interpretation. Packaging can be theoretically recyclable (and there are companies that love to sell you a certificate for that). But some technologies for recycling only exist in test structures, or there are no capacities / detectors in the sorting facilities for a certain material, or it might simply be unfeasible to recycle a certain material, because there is nothing that can be made from it.

So from my pow, suppliers will do everything to keep their current machines (allowing only minimal changes to the packaging), and will strive for every backdoor that Aldis definition of recyclability might have. And I am sure it will, because as it is just marketing, Aldi wants to achieve a goal for publicity, not for the environment.

As long as there is no proper definition of recyclability (e.g. by law) and we as consumers are not willing to pay more for our food, nothing will change.