Ugh! You don't even know what the photo vendor is doing on their end, then.
I use a professional EIZO monitor for image and video work. It's calibrated and profiled to D50. Which, quite bluntly, is what everyone should be using. Not the horrible 6500K default. D50 is the only color setting that comes as close as possible to the color we see on a sunny day at noon. It's what the entire printing industry uses. If you purchase a color viewing booth from a place like GTI Graphics, the bulbs and gray paint they use on the units is D50.
Yes, there's a purpose for explaining that first. 🙂
Here's what may be happening. The photo vendor may not have their monitors calibrated at all. And worse, they may be using an RGB profile for their monitor that has nothing to do with a monitor. Worst choice of all would be using sRGB as a monitor profile.
What happens then is everything looks incredibly vibrant. Almost day-glow. You'd think they'd notice that and pick some other canned profile. But if they choose anything with a short color range, like the ancient ColorMatch RGB, their monitor color is going to be WAY off.
So, they think they're sending you images with rich, bright colors, because that's what their monitor looks like. But you see more of what it really looks like. Dull and washed out.
I could be entirely wrong here since I don't know, and can't see their setup. But it sure sounds like they don't know how to manage color.
There's only a couple of ways P3 could end up being the tagged profile. One, they have Camera Raw set to use P3 as the viewing color space. Once the RAW image is opened into Photoshop and saved, that's the profile it gets. Or two, they're using a wide gamut space in Camera Raw, but once it's opened into Photoshop, they may be doing a Convert to Profile and converting the color to P3. Hard to say since it also depends on what the working RGB color space is set to. Is that set to P3? Lots of questions to ask your vendor.
Per your self answer (I missed your added response), RAW images don't have a profile. They can't. It's not an RGB image so cannot be assigned an RGB profile. So any thought that a monitor profile comes into play somehow can't happen. What you view in Camera Raw is determined by the working RGB space you want CR to handle the RAW color translation as. Pick sRGB, and colors will be dulled down. Change it to WideGamut RGB, open the same RAW image, and it will be much more saturated.
But yes, phones and tablets are a nuisance. Though I have to say with my iPhone 14 Pro, with the pro color option and RAW turned on, it takes photos with amazingly accurate color. Almost as good as the Nikon D800 I used to have.
And then there's the device itself. As I look at my previous postings with the images on my 6th gen iPad, it isn't very accurate. While the sRGB and monitor comparison are obviously different on the EIZO, they look identical on the iPad. The red point of the 3D profile within the tristimulus? I can't even see it against the red to the right of where it should be.
But that's also not unexpected. The iPad is using a small RGB space for the display. In the opposite effect of stretching color out, the iPad can't display the richest red beyond a certain point in Lab, so everything past the iPad's 255,0,0 point gets squished down to the same/nearest color it can display.
Basically, color management can drive you nuts. Even if you understand it very well and have things set up properly on your end, you have no control over the millions of monitors out there that aren't even close to accurate. What other people see is very unlikely to look like what you see.