I have a bunch of dirt cheap Zigbee temperature and humidity sensor from Aliexpress. They all use the famous AHT20 I2C sensor, which is pretty good, accurate, and also incredibly cheap.
I noticed these sensors are extremely slow. The one labeled "SdE" is the bathroom. When someone takes a shower, the humidity plot kinda moves a bit, but the temperature plot remains limp.

So I checked the sensor datasheet:

The enclosure is the complete opposite of these recommendations: a microscopic pinhole in front of the sensor, with the whole internal volume of the enclosure acting as a "dead air". Also the two AAA batteries add a lot of thermal inertia.
I drilled a few holes in the plastic so the AHT20 sensor would breathe a bit better.

Results:

Now when someone takes a shower and it gets steamy, the sensor picks it up very well, both on temperature and humidity. I can now use it to control the ventilation. I drilled all the other sensors too.
Conclusion:
Is it common for temperature sensors to have such a slow changing time
Yes
because of the plastic body which stays "cold" for a long time, even after being moving inside a building?
Yes, and the batteries have a lot of mass too.
Do consumer-grade/home-automation temperature sensors that can move from 3°C to 20°C in a couple of seconds / minutes exist?
Couple minutes, yes, no problem, just take the board out of the enclosure, or drill some holes, basically whatever you need to do to expose the sensor to the air.
Couple seconds, also yes, but the thermal inertia of the PCB will be a problem. You'll need a fan blowing on it.
Another issue is firmware: because it is battery powered, it will try to save power, and radio transmission uses a lot of power. So these sensors only report temperature when it has changed by "enough" degrees (depends on the brand) and after some amount of time. So don't expect second by second updates.
Some sensors can be reflashed with custom firmware, or other means used to tweak the refresh interval. Of course if you set it to transmit every 5 seconds battery life will be much shorter than if it transmits than every 5 minutes.
Here's Xiaomi LYWSD03MMC, a bluetooth sensor, which was reflased with Zigbee firmware (the chip supports both). Resolution is excellent, noise is low, and accuracy is not bad either. I got 10 of these for 26€ and they're all within 0.5°C. The button cell battery is both a blessing and a curse: it has low mass, so the sensor tracks faster, but with this update rate it won't last very long. The high resolution is nice, but since it transmits on every temperature change, it means it transmits a lot. After calibrating them, I will set them to update every few minutes instead of 10s.

A while ago I needed a fast air temperature sensor. It has to be as small and light as possible for low thermal inertia, but it also needs a large area to exchange heat with the air. I used a SMD diode soldered on thin copper foil, flappin'around in the breeze.