House of Blades doesn't have all rules, so I'm not sure it's in that book. Wyld Runner can of course change pets, but the pet that is changed from and the pet that is changed to both have to be the only pet that wyld runner can take. Universal rules is that pets are status items and therefore only available to leaders and champs. So Wyld Runner can't take giant rats. This was a direct response to this question: "Can i replace 3 Phelynx by three giants rats attached to a Escher Wyld Runner ?". Only leaders and champs can take pets. Wyld Runners are one of those unique exceptions in that they can take their own pets. This is also partially because that's how they come in the box to preserve wysiwyg, which GW cares about in some cases.
This is how it makes sense to me:
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As you can see Wyld Runner aka Prospect aka Juve (Specialist) only has Phelynx cat listed. Leader and champs got Phyrr cat listed. Together with the checkmark indicating they have access to Trading Post and can take status items, so can take any other pet too.
This was all totally straight forward before non-leader / non-champ fighters got exceptions and could take certain specific pets. Now there are a couple of exceptions here and there, Wyld Runners being one of them.