Well, either getting punched in the face real good will cause your mouth to bleed, or a badly broken nose can cause blood to drip into your throat/mouth, or, if you have a sucking chest or gut wound, you’ll likely either be coughing or vomiting up blood, that is internally bleeding into either your lungs or gut.

Hell, you can cough up blood from a bad enough blunt force injury to the chest or gut, especially if you’re a bit past your prime fighting age, I’ve seen it happen a few times during kumite bouts.

Its a thing that does actually happen from many kinds of injuries, and its pretty visually striking… makes sense as a visual metaphor to indicate somebody is massively fucked up.