There are not that many highly qualified embedded developers around. And lots of highly qualified desktop / mobile / server developers may not be interested in an embedded development job.
So it seems that what you showed in your interview was enough to convince them that you can do the job - or at least that you are the best candidate for an embedded development job that they are going to find.
If you are OK with doing embedded development for a much higher rate than your normal web development jobs, then go for it. Be prepared to do your best during the working day and lookup things you are missing in the evening, but that phase will be over soon, and the higher salary will stay.
Maybe you shouldn't mention in the company what happened, but then it is quite possible that the guy looking for a web developer saw your hobbies, and knew that someone else in the company is in dire need for an embedded developer, and passed you on knowing that he can find another web developer any time.
If you are smart and think you are up to it, then go for it. It's not as if you have no experience; you have experience with embedded development, and with development in general.