Currently, we cannot search for unique pets in the Pet Journal with either a built-in filter or using “unique” as a search term. Seeing as this search feature works for presumably every other pet property, it feels like an oversight that this isn’t a thing.
To clarify, “unique” as in pets/companions that you can have only 1 of, correct?
Have you tried the Rematch addon? It might have an option to filter unique, but I haven’t checked yet.
Alternatively, if there’s a big enough need/desire for a unique-filter, I can request that it be added to WarcraftPets’s “Filter Pets By…” page. Not ideal since you’d have to search outside of the game, but better than nothing.
I only mention the Rematch addon and WarcraftPets because we still don’t have breeds showing in-game, and have to resort to using addons for it. Makes me doubt that they’ll add another filter to the default Pet Journal if we can’t even get breeds 
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Rematch doesn’t have it as one of the regular filters, but you can script it easily:
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Filter -> Scripts -> New Script
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Give it a name and put this one line in the big textbox:
return petInfo.isUnique
- You can save it and use that as a regular filter now.
I badly want breed to be exposed in the in-game UI. Every patch and beta among the first things I type is to check if there’s any new returns in C_PetJournal.GetPetInfoByIndex. Still waiting. 
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Pet Journal Enhanced does, though, and you can save favorite filter combinations as named menu selections.