Why would Mike give the German engineers his real name?

I have a few possible theories:

1) Mike trusts the head guy. I don't know that any of the other workers overheard Mike's real full name being used, but I could be wrong.

2) Finding someone only by their name, even an uncommon name, is no easy thing. It's not like he shared his social security number, and I doubt in this world you'd be able to type Mike Ehrmantraut into google and find a hit that's our Mike. If you did, it might be his listed employer, and good luck finding him that way.

3) Mike has Gus to back him up. I'm sure the guys hired here are aware of the monumental amount of wealth and therefore power that Mike's boss must have.

4) Mike's a former cop, I dont think much scares him. Especially if he can stand up to Fring.

It is sloppy and out of character for Mike.

Bottom line, there is no reason to tell the guy his real name. They were talking in a bar; there are dozens of things to talk about that don’t involve giving his name. As an ex-cop, Mike should know how to carry a conversation while withholding things.

While there is no reason to tell his name, there are lots of reasons not to. Once he gives it to the head German, he should assume the other Germans will learn it. If one of those Germans gets caught for something else or develops a grudge, or just shoots his mouth off at the wrong time, then the authorities have a name and a general location to associate with what is obviously a secret underground facility for nefarious purposes (drug lab would be any cop’s first guess). The cops running a search on Mike Ehrmentrout in the Southwest, with a description and approximate age, wouldn’t have to sift through many DMV records to find their guy.

The fact that Mike later releases the rest of the German workers, all of who have good reason to hold a grudge against him (he killed their friend) is exactly the reason to make sure they know as little as possible about him. You’re right. This was dumb and very much out of character for the smart, deliberate, and careful character that Mike usually is.

I wanted to second ruffdove's answer and I need more space than allowed in a comment. As ruffdove says, it was indeed out of character.

Mike's surname is extremely unusual -- I took German in college and would have noticed someone with that name but in more than a few decades after taking German, I have never encountered that name. It would be extremely easy to find someone with that last name and no doubt they had other identifying information.

At first I thought, Mike is just trying to act like the whole endeavor is a normal project but clearly it is not. Moreover, they all know that Mike murdered their boss and friend. Even before that, they all also knew that it was almost certainly illegal -- I don't know if there was a point at which Mike tells them that what they are doing is to build a drug factory, but being engineers, it is not hard to figure out what it is going to be or at a minimum, why the thing is being built underground is a big hint that the facility is not legal, whatever it is.

Mike's remark when the van stop makes the stakes crystal clear: If we were going to kill you, you'd be dead already. Everyone knew that there was a high probability, once Ziegler had "disappeared" that they would be next. They had legitimate worries during that long and uncomfortable ride.

And you can bet that Fring and Mike discussed killing them all -- of all the murders that occurred, those would have been among the most justifiable if also the most cold-blooded.

I suspect the reason they were able to return to Germany is a simple one: If they killed one, like Kai, they would have had to kill them all. But if that had happened, the German ambassador, and I am not exaggerating, would have made a formal complaint about the deaths of, what was it, 8(?) engineers.

What Fring and Mike had to worry about is whether their threat, once these men were securely back in their own country, would be enough to keep every single one of them quiet. Without Ziegler's murder, perhaps the fact that they had willingly traveled to a foreign country to perform illegal work -- that the men knew it was illegal was in Fring's favor, that might well have kept them quiet. But the murder is something that they would all have to think about for the rest of their lives and I bet someday one or more of them would decide to go to the authorities about Ziegler's murder -- perhaps that is something the writers will go into further in a future episode.