Yes. There are many drinks that are labeled "milk" that don't contain milk proteins, but we now have "animal-free dairy milk", which does. I can absolutely see someone being confused as to whether a given product labeled as milk will trigger their allergies:

https://perfectday.com/blog/why-animal-free-dairy-still-cont...

The dairy industry has always fought the FDA on this, arguing that anything besides milk from a cow should not be allowed to be labeled as milk:

https://agfundernews.com/dairy-farmers-urge-fda-to-crack-dow...

https://www.nmpf.org/on-almonds-dont-lactate-anniversary-dai...

The matrix is big and will only continue to grow: lactose-free milk, animal-free dairy milk, almond milk, oat milk, strawberry milk, etc. Multiply that across milk-derived product analogues like butter and ice-cream and it becomes even more confusing. The meaning on the allergy label is quite specific!