In English, a pretty common expression is "to leave a bad taste in someone's mouth" and it means to give a bad impression or make someone feel bad or disgusted. I might say "That violent scene in Game of Thrones left a bad taste in my mouth."
If a person said something left a serious taste in someone's mouth, it sounds a bit strange to me (native English speaker). But I think it means it made a serious impression on someone.
Does that make sense?