Yes be concerned.
These are personal items, that are used on the mouth.
Better to be safe than sorry.
Explain to your MIL. Nicely.
Or, explain that PER your Pediatrician, that is what he/she advises.
And sure, tell your Husband. I'm sure, he'd be concerned about the health of his little girl too. Common sense.
It is either not telling your MIL for fear of her, or suppose your daughter gets canker sores/cold sores or whatever it is, on HER mouth.
AND, if your MIL gets cold-sore outbreaks every few months... maybe she should get it diagnosed. Properly. Maybe it is not just a "cold-sore" but something else? Who knows, right? So until then, don't share mouth products. Period.
And when/if your daughter is able to understand, you teach her about hygiene and about sharing personal products. Because it can spread germs. Even sharing Hair-Brushes... can spread head lice. Which kids in school, learn about.
All the best,