There's an example from Advanced Grammar in Use:
I remember the horse winning the race. (but not ... the horse's winning...)
I don't know why, but don't we say these as well:
I resented Andy's winning the gold.
Mickey recalled her buying the comics.
Then, why does the author make an exception for animals?
Let's consider the following sentences:
I resented Andy winning the gold.
I resented Andy's winning the gold.
In the first sentence, "Andy" is the direct object, and "winning" is a present participle. In the second sentence, "Andy's" is in the possessive case, so it can not function as a direct object. "Winning" is a gerund and functions as the direct object.
The sentences have slightly different meanings. In the first sentence, I resented "Andy" (who is described as "winning"); in the second sentence, I resented "winning" (which is described as "Andy's").
Both sentences are correct, but people will often write the first sentence when they mean the second. This is so common that many people think that it is fine, but strict teachers (and editors) will change it to the second sentence.
If you remember the horse, then you should write: "I remember the horse winning the race." If you remember the winning, then you should write: "I remember the horse's winning the race." (However, as I said, many people simply write the first sentence in either case.)
As cruthers noted in his comment, this has nothing to do with animals. It applies in general.
As cruthers also noted, the last sentence is ambiguous. "Her" can be a possessive pronoun (i.e., function as an adjective) or an object case pronoun (i.e., function as a direct object). Both versions of that sentence would be identical.
