Good question!
Just to clarify so everybody is on the same page: a skin biopsy is where we remove a full thickness piece of skin. In most cases we take a piece of abnormal skin, the junction where the skin transitions from abnormal to normal appearing, and a piece of normal skin. The biopsy is then fixed and sectioned by the pathologist, stained and then evaluated on a slide. The pathologist is evaluating the type of cells present, their organization and relationship to each other, and identifies if there are any abnormalities present.
How does this help us diagnose the problem? The cell population present can be very important in trying to figure out what disease process is happening in the skin. For example: differentiating between cancerous cells versus inflammatory cells, differentiating between different types of inflammatory cells, characterizing infectious agents present, etc.
The skin biopsy is not likely to tell you if you have a systemic allergy beyond telling us that there is significant inflammation within the skin, suggesting that the underlying issue is allergic skin disease. Similarly, we may see yeast or bacterial organisms in the skin on histopathology, but it doesn't tell us the type of bacteria or type of yeast present; we usually submit a skin sample for culture to characterize the type of bacteria present and define what antibiotic it is sensitive to.
Diagnosing specific allergies is typically done with intradermal allergy testing, the exact same way allergies are characterized in humans. This is a test typically performed by a specialist (as its expensive, and can be very difficult to interpret correctly) - a board certified dermatologist performs these tests at our hospital.
As for skin biopsies - are they important? The answer is definitely! They can be performed at your family veterinary clinic, usually with mild sedation and local anesthetic and they can provide invaluable information for your veterinarian to try and help your dog.