Neutral:
* Pet enthusiast: This is a straightforward, neutral term.
* Animal lover: This implies a strong affection for animals in general.
* Someone who wants a pet: This is simple and accurate.
Slightly humorous/ironic:
* Pet-obsessed: This suggests an unhealthy fixation on pets.
* Pet addict: This implies a dependence on having pets.
More specific:
* Someone with a "dog allergy" (or "cat allergy") - This is a humorous way to describe someone who always wants a specific type of pet, even though they might not be able to have one due to allergies.
It's important to consider the context and the tone you want to convey when choosing a term.