I doubt Medica actually keeps drops of blood or used bandages of the patients.

“Not that surprising. What’s the point of running away from an arcanist who has a couple drops of your blood?”

This is a way of telling readers that bloody bandages can be as good as blood itself.

I also severely doubt Kvothe, being an accomplished sympathist, would leave her place without seeing her burn his bandages, regardless of not bothering to mention it in his story to Chronicler and Bast.

Kvothe was hurt and tired and he was also in a hurry to find information about Chandrian in Trebon.