If the golden snitch was the school’s property, not that of Dumbledore, then how come he mentions the snitch in his will to Harry?
Used snitches seem to be given away by the school as prizes to the seeker who caught it. We see Harry with a Snitch (admittedly described as "stolen") in the common room after the game and is letting a cat play with it. Clearly it's not a high-value item after its 'flesh memory' has been activated or he'd be much more wary about letting people see him with it.
The Snitch he had caught earlier was now zooming around and around the common room; people were watching its progress as though hypnotized and Crookshanks was leaping from chair to chair, trying to catch it.
Harry Potter and the Order of the Phoenix
As Headmaster of Hogwarts, it's presumably within Dumbledore's gift to give himself something that would otherwise be considered junk.
Snitches remember who touched them. The Ministry had a clue that something might happen if Harry touches that particular snitch so even if it was school property, Scrimegour willingly gave it to Harry in the hope that it will reveal something that Dumbledore meant to show only to Harry. And once Harry had the snitch, it would be weird to take it back.
In 7th chapter of 7th book:
“I don’t think there’s anything hidden in the icing,” said Scrimgeour, “but a Snitch would be a very good hiding place for a small object. You know why, I’m sure?”
Harry shrugged. Hermione, however, answered: Harry thought that answering questions correctly was such a deeply ingrained habit she could not suppress the urge.
“Because Snitches have flesh memories,” she said.
“What?” said Harry and Ron together; both considered Hermione’s Quidditch knowledge negligible.
“Correct,” said Scrimgeour. “A Snitch is not touched by bare skin before it is released, not even by the maker, who wears gloves. It carries an enchantment by which it can identify the first human to lay hands upon it, in case of a disputed capture. This Snitch” — he held up the tiny golden ball — “will remember your touch, Potter. It occurs to me that Dumbledore, who had prodigious magical skill, whatever his other faults, might have enchanted this Snitch so that it will open only for you.”
The answer probably is concerned with 'how can Dumbledore give the snitch' in his will as he obviously has 'given' it.
Recall how he also tried to give Harry the sword but this was blocked by the Ministry (mainly because they didn't know where the sword was).
However, with a snitch, it may well have been allowed but the Ministry could also have blocked it - why didn't they ?
Everyone knows that snitches have 'flesh memory' and a snitch in and of itself is pretty useless outside of a Quidditch match. The Ministry were therefore asking why did Dumbledore bequeath a snitch and they came to the conclusion that there was more to it than meets the eye. Could it be a message from Dumbledore to Harry or could it conceal something useful. Remember Dumbledore didn't let the Ministry in on his plans which left them somewhat 'put out'.
So, I suspect that the Ministry let it pass so, when the snitch was given over to Harry (by the Minister of Magic himself), something of Dumbledore's plans would be revealed to them via flesh memory.
As we know this never happened (due to how Harry caught the first snitch and 'opens at the close') and the Ministers reaction of disappointment when nothing happens speaks volumes...