May be you have a problem with you monitor(!) profile or the configuration of Chrome/Firefox.
What I tried:
1. I created an image in Photoshop (profile sRGB, color #4429, RGB 206/68/41).
2. I saved this picture as JPG.
3. I placed it in Muse.
4. I used the color picker to show the color values – exactly the same as in Photoshop!
5. I exported the site as HTML, took the exported picture from the "images" folder and opened it in Photoshop: exactly the same colors!
6. I made a screenshot of the Muse layout containing the picture, opened it in Photoshop (sRGB): exactly the same color!
7. I opened the exported HTML with Safari, saved the picture and opened it in Photoshop: exactly the same color!
8. I created a box in Muse using the same colors as the image, made a screenshot opened it in Photoshop (sRGB) and you certainly know what happens: exactly the same colors!
Whatever I did, the color remains the very same.
So the answer to your headline "Why can’t Muse and Photoshop agree on colors?" is obvious: They DO agree!