Shards limits are entirely arbitrary and not representative of what the shards can actually do (Taylor is the best example since we know that QA is fully capable of controlling any organism, with specific examples including rats, birds and, of course, people, and also its definition of a bug is mostly based on vibes). They're also capable of observing the world to some degree and making adjustments when necessary. Taylor is once again our example since she's gained her multitasking after initially being overwhelmed by the sensory input, and QA adjusting her power to be more useful.
So, when it comes to Rachel and Whisper, it's less a matter of whether her shard can empower Whisper and more whether it'd want to make an exception for this person based on thematic connection. The answer is probably no, shards do tend to stick to their self-imposed limits outside of second triggers and impromptu lobotomy. Unless you think it would lead to a cool story, in which case the answer is yes because shards' motivations are vague enough that you can justify it.
As for the reverse, well, it's not like Rachel is literally a dog, she just understands them very well and doesn't understand humans very well (a problem implied to exist before her trigger), so any form of direct control probably won't work unless it's some magic operating on thematic connections or somesuch. If it's more about clarity of communication, then Whisper could probably make herself understood very well by Rachel, better than anyone else around, which could easily foster a positive relationship.