There’s one more reason we haven’t really mentioned yet. If we assume that leaving the ID is not an individual choice, but rather one of the instructions by the IS, it obviously has a lot of advantages.
There is nothing to be gained, for the IS, from the attacker remaining incognito. There is, maybe, a slim chance that he’d stay free and have the opportunity to do it again, but in the end, that is very unlikely and risky. On the other hand, showing everybody that he came to Germany as a refugee? As closely as possible to the attack itself? Yeah, that’ll fuck the situation right up.
Predictable response is the CSU and AfD blaming all refugees and Merkel for the attack, general xenophobia rising and anti-refugee and anti-muslim hatred and violence rising. Thus making recruitment easier for the IS.