Lightroom Color Profile Changing from Adobe Color to "Color" after saving file in Photoshop

A raw file from a digital camera and an RGB file from Photoshop are two very different things. That's why we need raw processors like Lightroom/ACR. Photoshop cannot work directly with raw files, it needs to be processed into an RGB file first.

"Adobe Color" is a camera profile. It is used in the processing of raw files from a camera. When the file has been processed in Lightroom and opens into Photoshop as an RGB file, the camera profile (Adobe Color) has already been baked into the data. It has served its purpose and done what it was supposed to do. It no longer applies and can not be used on RGB data.

Camera profiles should not be confused with icc profiles like sRGB, Adobe RGB etc. They serve very different purposes.

Adjustments in Lightroom will have different effects on raw files vs. RGB files. They will look different and to achieve the same visual effect you may need different adjustments. You cannot treat raw files and RGB files the same way, and you should always keep this distinction in the back of your head.

I would also recommend that you do not use ProPhoto until you get more experience. ProPhoto can easily do more harm than good if you're not fully aware of all the implications. Just change the Lightroom preferences for External Editing into sRGB or Adobe RGB. This profile will be preserved into Photoshop and override whatever is set as working space in Photoshop.

Lightroom and Photoshop color settings do not need to match. That's a common misunderstanding. Any incoming profile will be preserved and correctly handled in Photoshop.