animal hoarding
pp. Keeping more pets than one is able to adequately care for.
Other Forms
Harc defines an animal hoarder as someone who accumulates large numbers of animals; fails to provide basic nutritional and medical care; and, crucially, does not recognise the negative impact on their own well-being. Hoarders typically believe that they, and they alone, can help these animals. Many hoarders are in a state of denial and won't accept that they live in squalor or that their animals are dead or dying. According to Patronek's research, the profile of hoarders often fits the stereotype: 76% are female, 46% are over 60 and more than half live alone. That said, there are also cases in which much younger people, couples, working professionals and even a few vets were found to be hoarders.

Despite the seriousness of the problem, research on animal hoarding is in its infancy. We don't really know why people do it or what should be done to tackle it. Harc researchers are just beginning to plug the gap. They have suggested that hoarding might be a symptom of delusional or attachment disorders, dementia or obsessive compulsive disorder.
—Justine Hankins, “Love is the drug,” The Guardian (London), March 01, 2003
1998 (earliest)
He said cases of what he called "animal hoarding" are not uncommon. Sometimes, he said, the cats have litters of kittens and the owners can't imagine that anyone else could care for the animals.

"I don't think it's intentional abuse," Fennessy said "It just snowballs."
—“Officials remove 100 cats from Northampton home,” The Associated Press, October 05, 1998
The "Harc" mentioned in the example citation is the Hoarding of Animals Research Consortium, which is part of the Center for Animals and Public Policy at Tufts University in Massachusetts. HARC was formed in 1997 so — despite the date on the earliest citation I could find — it's possible the phrase animal hoarding dates from that time. Note that a synonym — animal collecting — dates to 1994.