Can the post-processing of video improve the color of characters in the video like PS, and achieve the effect of beautifying the picture?

Yes, but it is time-consuming and laborious. Adobe's After Effect (AE for short) can solve your problem, and its working principle is similar to that of PS, except that PS creates layers on static pictures to modify, and AE creates layers on dynamic videos to modify. In fact, video is made up of frames of pictures. To modify the color or face of a character, you must first know the moving path of the character in the video. To put it bluntly, it is what actions the characters' faces do in the video, and according to these actions, key frames are established and each key frame is modified. So your problem can be solved, but it is very laborious. Like the characters in movies, they are all modified, otherwise they won't look so good.