No.

That makes no sense. In every camper I've ever seen has a plumbing set up roughly simplified per the below diagram. The city inlet goes directly to the "cold" line throughout the coach, and the fresh water fill port goes to a tank. The bottom of the tank goes to a pump to the same "cold" line throughout the coach.
(Fun fact: one of the most common complaints is "why does my fresh tank overflow when I'm on city water supply?" This happens because the system is SO simplified that it relies on the pump being a check-valve preventing the city inlet from back-flowing to the tank. As the pump wears with use, it can leak, and a very common piece of advice you'll see here is installing a separate check valve to stop this leak.)
Forgive me if I'm patronizing, but are you new to RV ownership? A very common misunderstanding is how tiny the water heaters in these coaches are, and how under-powered their electric elements are compared to running the water heater on propane. It's really easy to see how if you're only on your third or fourth trip, you're just hitting bad timing here. A 5-gal HWH is typically only good for a 3-5 min shower. We've got a busy family of 7 and a very common pattern for us is:
1. I shower. 2 mins, in and out. Plenty of hot water.
2. My wife showers. 3 mins, she has a lot more hair to wash.
3. Turn on the sink and hot water comes out - but the tank is JUST about done now. The HWH is trying to recover.
4. On electric only that HWH needs maybe 20-30min to fully recover. On propane it needs maybe 8-10. (Every unit is different, that's just our experience with ours.)
5. I wash my hands. Hot water.
6. Kid goes to shower. "Dad, the hot water heater is broken again!"
Now mix up the above schedule and factor in a few negatives. Most RV pumps are "set" to maintain about 30-35psi. City is SUPPOSED to be that but if you hook a pressure gauge up you'll find it's very common to see these run 40+. I've even seen 60+ at some campgrounds. (Fun fact: another common failure mode in these campers is having your "city" supply be so high that it damages your plumbing, and another very common item folks add to their rigs is a "pressure regulator" which is why there are like 2 dozen options for that on Amazon.)
Now add in the fact that a typical RV bathroom faucet is going to run like 0.5gpm but the shower heads most rigs come with are probably 3-5. (Fun fact: a very popular upgrade is the "Oxygenics" shower head, which pulses the water so you feel like you're getting the same pressure but at <2gpm.) It's really easy to see how you can turn on your bathroom or kitchen faucet and get water water, but your shower runs out in 10 seconds and it seems like you have no hot water.
If you want to confirm all this, repeat your test but wait like 4 hours first and then test ONLY the shower. I would bet you get hot water then. If not, I have literally no idea how you could make just one appliance NOT get hot water without installing a lot of extra valves and sensors (to detect which supply you're on and control those valves) to MAKE that occur, and will shut up and defer to other brighter minds here.
