My dog coughed mucus and vomited. Why didn't the vet give him meds?

Hello and welcome to Petco Pet Education Center, formerly Petcoach.

Coughing can be caused by many different conditions including heart disease, collapsing trachea, infectious tracheitis (kennel cough) and pneumonia (fluid in the lungs).

I would be concerned about a bulldog coughing after known vomiting, as they are at a higher risk of aspirating (fluid from the stomach going down the trachea and into the lungs instead of coming out the mouth or back into the stomach).
I would recommend having Valentino rechecked, and discuss x-rays with your vet to see if there are signs of aspiration pneumonia.

However, if Valentino is only coughing a tiny amount, and is otherwise normal, eating/drinking, playing, running around normally, then he likely doesn't need antibiotics as some aspiration pneumonias are sterile (no bacteria present).

If the coughing is continuous, or any continued vomiting, or if Valentino is lethargic, weak, not eating, and not acting himself, or has any signs of trouble breathing, then get him to a vet immediately.

Best of luck with Valentino.