Many years ago French ATC checked I was VMC à propos of nothing, which I found a bit intrusive as it wasn’t under a radar service, or for purposes of a PIREP. If you are declaring a PAN or Mayday then advising or being asked whether you are able to fly IFR is conventional.
I would regard that Q as innocent, and is often used if you ask for a shortcut which cuts a corner off their MRVA, and they don’t want the form filling if you hit some granite 
There is a particular route across the Alps (runs N from SRN) on which I always push them for a shortcut, and I get that Q there.
But I do enjoy “do you have oxygen” 
However, nothing beats the “how iz dat possible in a TB20” from a nice friendly Spanish ATCO
I once heard (in the US) “do you have a map?”, and replying in the affirmative, the feckless pilot received instructions on how to use it!
In the summer here with lots of foreigners flying VFR or IFR or whatever they do, I hear from time to time from the tower: “[callsign YYYYY] What are you doing??”
Most questions ATC ask are there to give them information on how to handle you. Much faster than calling up the flight plan and dechiper the equipment box.
Obviously when you start to display erroneous behaviour they might ask a bit more pointedly.
But be glad that ATC is still a person to person talking exchange. I’d miss it if it wasn’t. Talking to them is fun and mostly a very happy thing, at least for me.
My favorite bonmot was overheard once near Lyon… When asked about his postion (looooong before GPS) the pilot of a Cessna replied “over the Rhone?” to which the controller replied “Err, ok, you know the Rhone is kind of long, can you be a bit more precise?” The situation was finally sorted when the pilot managed to activate his transponder and was found to be over the Saone north of Lyon as opposed to South over the Rhone… Oh the beauty of GPS…
Not really on topic, but my best ATC interaction along these lines was flying VFR from Palo Alto to Lake Tahoe.
The controller asked, “Do you have the train in sight?”
I started looking for the railway line which I knew as somewhere underneath me, which goes through the Donner Pass of evil repute. Then I realised that probably what he meant was “terrain”. Which indeed I did.
Interesting thread, with the most notable thing being that (as usual) not a single ATCO has posted anything Surely they must know the rules – for their country at least.
It has happened a few times when flying longer distances above 5000 ish feet, the ATC asks if I’m flying IFR. My answer is always no Once in Sweden, 5-10 minutes before landing after being in contact with Sweden control for about an hour, the next ATC asks: Do you have a transponder? I became a bit confused, since the transponder had been on all the time, and still was on with the correct squawk. I just answered Affirm. Then she said: would you mind turning it on? After some more short messages, I turned the transponder off then on again, and she could see me. A Trig Transponder. Have seen the problem later also, intermittent, and it sometimes shows up on the display that transmitting power is low, or something. Sometimes re-booting it helps, sometimes it doesn’t. Haven’t really investigated, but perhaps some antenna/power issue, poor connection ? I’m sure the ATC has another cause for this problem in their minds
In the US, ATC doesn’t have easy access to the equipment one filed. The original FAA domestic filing system was based on single character slash codes for equipment, like /G means a mode A/C transponder and GPS enroute/terminal capability. /U is VOR, no DME, with A/C transponder. /L is RVSM, GPS capable, Mode A/C transponder. So when ICAO equipment codes came along, they were translated in the ATC software to their equivalent slash code that is displayed to the controller. I understand that the centers will in the near future (years?) get away from the slash codes, but the terminal controllers systems will still be using them. That means that the controller doesn’t know that you are capable of flying an RNAV (GPS) approach, much less that you can use LPV minimums. So there are questions asked in the normal course of events regarding equipment.
I haven’t flown much GA in contqct with ATC but the once they did ask for my capabilities.
On the 1st leg of a planned 7-10 day trip around France but the weather window was very small to depart and of course we didn’t manage to leave on time.
We already turned back to our home airfield but i didn’t like the terrain and the low clouds so i decided to clinb into the clouds and to divert to the big controlled airport.
The controller asked if i was able to fly IFR. And I didn’t lie when I said I was rated on the 737.
The plane didn’t havr his IFR check and my IR is only valid in multi engine, multi pilot operations.
At that moment the question didn’t bother me and it actually made me feel safer they were looking out for me.
We were also the only plane on Brussels info that day. If we could have left on time we would have had an awesome trip
