How do you explain this feeling? Is there a word for it, even?
It's like... needy. The monkey needs a hug.
It's not love, it's not even necessarily affection, because those words imply caring for the other—and the way the monkey is staring fixedly past the puppy creates a sense that this might be a one-way relationship at the moment, that the monkey just needs this hug and maybe the puppy is along for the ride.
What's the relationship here?
Of course, we're reading a lot from a moment frozen in time. Maybe one second later the monkey turns its head and nuzzles the puppy affectionately and the puppy licks back.
Maybe the puppy sought out the monkey for a hug, and the monkey is the one going "sure, I guess I'll give you a hug"?
Whatever the case, the feeling we can empathise with is needing a hug.
Interspecies empathy
Do you notice how we feel empathy even when the subject or character isn't human?
We have a good ability to read emotions in other species (and some animals—such as dogs and apes—can read our emotions in turn).
This ability to anthropomorphise, or see animals and objects as having human traits, is also what lets us create stories that feature them as characters.