"Full caster" is not a distinction made in the rules.
The question seems to be built on an incorrect assumption. The rules do not define "full caster", and consequently, there are no requirements based on the concept. Some magic items and feats require you to have a spellcasting class feature or to be a spellcaster, but none of them mention the concept of a "full caster".
This idea comes from the multiclassing rules for determining your available spell slots:
You determine your available spell slots by adding together all your levels in the bard, cleric, druid, sorcerer, and wizard classes, half your levels (rounded down) in the paladin and ranger classes, and a third of your fighter or rogue levels (rounded down) if you have the Eldritch Knight or the Arcane Trickster feature.
"Full caster", "half caster", and "one third caster" are terms the community came up with based on the spell slot progression described here, but the rules don't actually use these names to refer to these classes anywhere. In this context, classes and subclasses are categorized based on the number of spell slots they contribute, with Bards, Clerics, Druids, Sorcerers, and Wizards contributing the full array of spell slots, and so are sometimes referred to as "full casters" (by players, not the rules); Paladins and Rangers contribute approximately half that of a full caster, and Eldritch Knights and Arcane Tricksters have approximately one third the available slots of a so called full caster.
So to make a magic item that requires attunement by a "full caster", the rules-consistent way to write that would be something like:
Requires attunement by a druid, bard, wizard, cleric, or sorcerer
Written this way, multiclassing would present no complications for the item's use, as having even one level in one of these classes is sufficient to meet the requirement.
