Yes, it is possible to catch thrush from your baby while breastfeeding.
If your baby has oral thrush then the regular contact between yourself and your baby means that you could also develop a yeast infection in your breasts or nipples.
This is why your doctor may also want to test your breasts for thrush if your infant has a yeast infection in their mouth.
It’s not unusual for both a breastfeeding mother and their baby to have thrush at the same time.