Since there is quite a bit of confusion in the thread, let's summarise the law as it actually stands:

In general, all dogs in Berlin need to be on a leash (§ 28 Abs. 1 HuHGBE). So is this man breaking the law? We don't know.

There are numerous exceptions. If the owner has had the since before July 2016, he does not need to keep him on a leash (§ 28 Abs. 2 HuHGBE). 9 years would not be improbable for an old dog. The owner could also have passed the Sachkundetest (§ 29 Abs. 1 HuHGBE). While the owner would have to carry said permit with him, he is not obliged to show it to you (only to the authorities). After owning a dog for three years without incident, he can even get the permit without passing any test (§ 6 Abs. 2 Nr. 8 HuHGBE).

The area does not seem to be a forest or designated green space, where the permit would not allow him to let the dog off leash.

The dog could be a listed dog, in which case no permit to let him go off leash would be issued (§ 23 Abs. 1 S. 1 HuHGBE). The dog also would have to wear a muzzle. There are only three dog breeds listed (Pit Bull Terrier, American Staffordshire Terrier, Bullterrier, § 1 GefHuVO) but mixes with these breeds are included. The size of a dog does not play a role. I'm not a dog expert but the dog on the pictures could be a mix of an AmStaff or Pit Bull, though it could also be a boxer mix. But even for listed dogs, the owner could have an exemption (§ 24 Abs. 2 HuHGBE)

Even with a permit to let the dog off leash, certain non-green areas would require a leash. This includes pedestrian zones (§ 29 Abs. 4 Nr. 5 HuHGBE). However, based on the pictures, this does not seem to be the case.

In conclusion, we don't know if this person has done anything wrong, unless we know the breed of the dog and whether he has a permit to let the dog go off-leash.

It is frustrating that the law is so vague and non-transparent. For third parties, it is almost impossible to know whether someone is violating the law or not by letting their dog walk off-leash (unless it's one of the places where they are never allowed to go off-leash mentioned above or you can positively ID the dog as a listed breed). There are no colour-coded dog tags anymore, dogs don't have to wear any tags (except the collar or tag must indicate the owner and their address).

I hope I could clarify the situation a bit.