A name for one specific sort of pathological "stuff-collecting" is (since 1975):
Diogenes syndrome, also known as senile squalor syndrome, is a disorder characterized by extreme self-neglect, domestic squalor, social withdrawal, apathy, compulsive hoarding of garbage or animals, plus lack of shame. Sufferers may also display symptoms of catatonia.
The condition was first recognized in 1966 and designated Diogenes syndrome by Clark et al. The name derives from Diogenes of Sinope, an ancient Greek philosopher, a Cynic and an ultimate minimalist, who allegedly lived in a large jar in Athens. --https://en.wikipedia.org/wiki/Diogenes_syndrome
The eponym is a bit tongue-in-cheek: Diogenes of Sinope was an ascetic, and adopted the decision to carry nothing with him that he did not need.
Here is an example of his minimalism (emphasis added):
The stories told of Diogenes illustrate the logical consistency of his character. He inured himself to the weather by living in a clay wine jar belonging to the temple of Cybele. He destroyed the single wooden bowl he possessed on seeing a peasant boy drink from the hollow of his hands. He then exclaimed: "Fool that I am, to have been carrying superfluous baggage all this time!" --https://en.wikipedia.org/wiki/Diogenes
People with the syndrome may have millions of things that they actually can't ever need, but usually rationalize their collection "because I need it", or perhaps "because I will need it someday", thus saying the same thing about their hoards as what Diogenes said about his non-accumulation of anything. The humor is fitting, though: Diogenes was a cynic.
Another name for Diogenes Syndrome, syllogomania, is mentioned by Elizabeth Andersen et al. in a case study at https://www.researchgate.net/publication/255950905_Do_they_think_I_am_a_pack_rat. In this same paper, the person who hoards is called a "hoarder", as stated in a previous answer. The opening paragraphs express the pathological nature of the condition referred to in your question (emphasis added):
Hoarding is a debilitating disorder characterized by the acquisition of a large volume of possessions that seem useless or of little value to others (Frost & Gross, 1993; Stein, Seedat, & Potocnik, 1999; teketee, Frost, & Kim, 2001). The compulsion to acquire items can become sufﬁciently severe to cause marked impairment in activities of daily living, social activities, and relationships with others. Broad acquisition of free things is often accompanied by compulsive buying (Frost et al., 1998).
Hoarding of rubbish (syllogomania) is a symptom of a much broader syndrome of senile self-neglect or squalor (lack of personal cleanliness and hygiene, domestic squalor, hoarding, rejecting society, resisting offers of help, and lack of shame). Initially labeled as “Senile Breakdown” (MacMillan & Shaw, 1966, p. 1037), the syndrome was relabeled as “Diogenes Syndrome” (Clark, Mankikar, and Gray, 1975, p. 366) after the Greek philosopher Diogenes, who pursued self-sufﬁciency, rejected luxury and domestic comfort, and freed himself from material possessions. (...) Their living spaces are usually taken up with stored garbage, urine and feces in buckets or in packages, and masses of hoarded material, which is the antithesis of Diogenes’s freedom from material possessions.
"Syllogomania" is also mentioned in A Dictionary of Neurological Signs by A. J. Larner (here).
In keeping with other "-manias", then, the hoarder could be called a syllogomaniac, but at the time (2020) a quick Google search shows that word only in a twitter hashtag: https://twitter.com/hashtag/syllogomaniac.
Another label option is disposophobe or disposophobic whose condition is called "disposophobia" at https://www.fearof.net/fear-of-getting-rid-of-stuff-phobia-disposophobia/.
An unverified example at https://glosbe.com/en/en/disposophobic includes:
A disposophobic has a delusional regard for the value of the things he hoards.