Many people who use whose (or whom) rather than that (or which) when talking about pets or anything to which they ascribe emotions, animus, and other kinds of "personhood." (That's certainly true of the student's comparison of her puppies to anyone else.)
Not only is it commonly acceptable to use who, whose, or whom with pets, but whose can also be used with inanimate objects.
From Mignon Fogarty's "Pronouns for People and Animals: "Who" or "That"?":
The Associated Press provides useful guidance about animals, presumably because their writers find themselves writing about pets often enough to need clarification. The AP Stylebook recommends using the pronouns “it” and “that” for animals unless you know the animal’s sex or the animal has a name.
For example, if you were writing about a cat and you didn’t know much about it, you’d write something like “The cat that was stuck in the tree peed on the firefighters.”
If the cat had a name, you’d write something like “Fluffy, the cat who was stuck in the tree, peed on the firefighters.”
If you only knew that it was a female or you were just using pronouns, you’d write something like “The cat who was stuck in the tree meowed at her owner.”
In contrast, however, she does mention that other style guides give different guidance:
For example, APA style has different recommendations. They want you to always use “that” when you refer to animals such as cats and rats, not “who.” This is just a guess, but I wonder whether the APA might stick with “that” for animals because, unlike Associated Press writers, APA writers are most often writing about lab animals instead of pets. Regardless, it’s important to remember that even though the general rules are flexible, the rules may be more rigid if you’re following a specific style guide, so be sure to check.
Finally, she goes on to talk about inanimate objects:
Occasionally, people think or have been taught that they can’t use “whose” for objects, but the style guides are clear that it’s fine. In fact, Merriam-Webster says, “The notion that ‘whose’ may not properly be used of anything except persons is a superstition.”
If it bothers you though, you can always rewrite the sentence. “The table whose legs were broken” can become something like “the table with the broken legs” or “the table that had broken legs.”
I find that last example particularly interesting:
The table whose legs were broken.
I've seen and used constructions like this on multiple occasions and haven't observed it to be a problem.
More on this is said in "Whosetrionics" at Grammarphobia:
“Whose” was first recorded in the late 800s, when it was written in Old English as hwæs, according to the Oxford English Dictionary. In those times, it was the genitive case of both “who” (hwa) and “what” (hwæt), the OED says.
During the Middle English period (roughly 1100-1400), the spelling of “whose” shifted a lot, from hwas to the later hwos and whos. During this same period—in the 1300s—people began using “whose” as the genitive form of “which” as well as “who.”
. . . In A Dictionary of Modern English Usage (1926), Henry Fowler vigorously condemned the taboo against using “whose” for things:
“Let us, in the name of common sense, prohibit the prohibition of whose inanimate; good writing is surely difficult enough without the forbidding of things that have historical grammar, & present intelligibility, & obvious convenience, on their side, & lack only—starch.”
The current Fowler’s Modern English Usage (rev. 3rd ed.), by R. W. Burchfield, says the notion that “whose” is limited to people is a “folk-belief.”
So, the only real prohibition against using these words is not one of grammar per se, but one of personal preference and the particular style followed. There is no universal "correct" way, only the way that you follow.
