People screw this up all the time. They think that a 1080p stream and a 4K stream are just a jump in resolution.
I have news for you - your 1080p stream is NOWHERE NEAR 1080p.
It's all about video compression that is in use and the bit rate that they are using. You see, a 4K Blu-ray Disc may look better on ANY display because of the newer codecs in use and the much higher bitrate which is used for the encoding. Whether that video had more detail or not to begin with, the encode itself is of higher quality than the 1080p version of the same movie, and almost any display can take advantage of that. As well, the expanded color pallet, which is not the same as HDR from my understanding, also can come into play. So, you can get greater bit depth in the colors without necessarily getting HDR content, and that can impact image quality with 4K encodes.
Now, you get into streaming, which is really a joke in comparison to Blu-ray Disc quality. You should be able to see artifacts at 1080p that should never be there. I certainly watch streams on my projector (1080p) and they look like garbage compared to Blu-ray Disc which is typically razor sharp. HDTV often looks like crap as well. So, I can certainly see that switching to a 4K stream, at a much higher bit-rate will eliminate a lot of the encoding issues that they don't care about with their 1080p versions.
This isn't about the encode, it's about the added feature set that is included across all 4K material, whether your TV accepts HDR or not, the added color depth is on the disc, and this comes across on the TV. The higher bit rate of the encoding is there, and it comes across on the TV. If you had a 1080p TV that had the ability to receive a 4K signal and the expanded color gamut, then it may very well look identical to the 4K TV from the same seating distance. But, that's not what you are testing. You are starting with two extremely different encodes, and you should always expect to see a difference.
You may even see a jump in quality if you went from a 4K movie that is being streamed to a 1080p Blu-ray Disc copy of the exact same movie. The disc may look better overall even though it should have a lower color gamut to begin with and lower resolution claim, it does it all with a much higher bit rate of encoding.