No, it's not possible.
A true allergy is mediated by histamine (released by the mast cells) and involves an "immune system-mediated adverse reaction to food proteins." (Or to non-food proteins in the case of other allergies.) While many sources assume IgE is the only type of true allergy, it can also be IgG, IgA, and others.
Water doesn't have proteins, unless they come from contamination (or from being in a natural state such as a creek or water faucet or in blood).
A pseudo-allergy is a very real condition where something causes release of histamine (or
histamine can come from external sources) and the end result is very much like an allergy. Aquagenic Urticaria could be classified this way.
Then we have intolerance (food intolerance or others). Allergies and pseudo-allergies are types of intolerance. This simply means (the inclusion of) some medical reaction that isn't mediated by histamine. Lactose intolerance and celiac disease are examples.
Your character might be intolerant of water (something that a layperson could easily call an allergy) but, as others have pointed out, that doesn't work. Water is most of our bodies and ubiquitous.
I've known people in real life who were "intolerant of water" but it's always about contaminants in water. In the case of people with extreme reactions, it's generally about plastic. But you've ruled this out as a possibility.
So what you're left with is a situation that isn't real but can still be part of a fantasy story. This idea that somehow if a cow drinks water and produces milk, that renders the water safe for your character. Or if an orange tree produces fruit after rainfall, somehow the tree is able to convert the water into a safe juice (I'll assume that your character doesn't drink the majority of orange juice brands which are orange juice concentrate reconstituted with water).
The novel Wicked does this. It's a take on the Wizard of Oz story, told from the point of view of Elphaba, the Wicked Witch of the West. As anyone who has read the books or seen the movie knows, Dorothy kills the witch by pouring water on her, causing her to dissolve. In this novel, Elphaba has a lifelong reaction to water. She bathes herself by rubbing her skin with oil and scraping/wiping it off (also done in some ancient cultures).
[Book version] Elphaba is also unusual in that she is apparently allergic to water,
and avoids touching it at all times, never crying or bathing. She
cleans herself by rubbing oil into her skin. She has a power that she
cannot control. This shows mainly when she is angered. An example is
when she sees Chistery trapped on an island in the middle of a lake.
Ignoring her allergy to water, she jumps into the lake to save the
monkey. However, when Elphaba touches the water it turns to ice for
her...the weather changed to suit Elphaba's needs. (ref)
So if you're looking for a scientific explanation for an "allergy" to pure unadulterated water in someone who not only lives but can drink milk and orange juice, you will be disappointed. But if you want to write fantasy where a character can not contact or consume pure water, or substances to which pure water has been added, you can do as you like.