Verb for refusing to share an item

Solution 1:

Verb for refusing to share an item

• monopolize

/məˈnɒpəlʌɪz/

verb gerund or present participle: monopolising

(of an organization or group) obtain exclusive possession or control of
(a trade, commodity, or service).
"they instituted press censorship and monopolized the means of communication"

^{Oxford dictionary: web}

If you wanted a less formal word in moderatley common use:

• hog

informal take or use most or all of (something) in an unfair or selfish way. "he never hogged the limelight"

^{Oxford dictionary: web}

Solution 2:

In addition to Bitter dreggs answer, there's a slang version that is not as popular as it once was.

Bogart

https://en.wiktionary.org/wiki/bogart

• To selfishly take or keep something; to hog; especially to hold a joint (marijuana) dangling between the lips instead of passing it on.

This expression was made popular by the band The Fraternity of Man (but was used before this):

https://en.wikipedia.org/wiki/Fraternity_of_Man

Solution 3:

Can the same be used for a single item? It doesn't sound right.

No. You hoard a quantity of something. It is not possible to "hoard" a single item.

The verb "to hoard" derives from the noun "hoard": 1.a. An accumulation or collection of anything valuable hidden away or laid by for preservation or future use; a stock, store, esp. of money; a treasure."