It's not much of a stretch, really. Not compared to slots, anyway. The idea of a magical artifact that will only work for a certain character, or which needs to be 'mastered' to use it is common enough.
5e's attunement is pretty nearly glossed over, but you could explore it in more detail, either in terms of fluff (describing the details of and reasons for the process in more detail), or crunch (requiring certain checks to successfully attune an item, for instance, or making un-attuning a thing).
Why 3? 3 is a fairly significant mystical number. Why do 'bad things come in threes?' Why do Christians have the Holy Trinity, and pagans the Mother, Maiden & Crone? D&D's been using it since 3/day limitations.
Depending on how common items are in your campaign, you could change it, too.
Limit 1: Items that require attunement are 'jealous,' and won't work together; it takes all one's force of personality to keep a single one in line.
Limit 2: Items that require attunement must be brought into accord with the psyche of the wielder. One such item is simple. A second requires the wielder to attune to each, and the items to eachother, in essence the wielder plays the two items off against eachother, keeping them functioning in his service. More than two would be able to 'gang up' against the wielder and overcome his force of personality.
Limit 3: A mortal can attune one item to his body, one to his mind, and a third to his soul. (For added interest, specific items may not be able to attune to one of those, or only to one of them.)
Variable limit: Attuning an item is a function of force of personality (CHA) or mental discipline (WIS). The stronger your personality/greater your discipline (higher your bonus) the more items you can attune (add bonus to the number of items you can attune, starting at 1 or 0 or something else depending on the campaign).
