It seems like part of what makes it _feel_ real is your brain's own ability to rationalize and explain what it's "seeing" under the assumption that it is real.
For example, some say you should try to look at a complex object like a watch or a phone since your brain will often get the details glaringly incorrect. However, I find my brain can often "explain away" exactly _why_ I don't have my phone with me at the moment, or why the watch that I'm suddenly wearing isn't accurate. Sometimes the dream then shifts to be about solving that problem - e.g. trying to retrace my steps to find my phone.
Your example of a dream problem scenario was amazing, by the way.
My dreams are always a mishmash of real and imagined places, scenarios and people. For example studying for exams but in a completely different city or country, stuff like that. My mind just accepts it, never seems to have any issues with it. I only realize the craziness after I wake up (if I remember the dream that is).
Your example of a dream problem scenario was amazing, by the way.