I don't believe there's a specific ruling on this ability, but here's my attempt. For the Belt of Many Pockets:
Wearer benefits should be distinguished from abilities of the item. Anything that requires physical interaction should be excluded from transference.
Its storage ability isn't a benefit conferred to the wearer, but an ability of the item itself. Thus, the belt should respond normally to the desire to retrieve an item (i.e. the relevant pocket opens), but it shouldn't be retrievable unless physically there with the belt.
For your question about spellcasting items:
Generally speaking, spell effects should originate from the item, not the caster. But there are exceptions depending on wording.
I say this going off the section on Using Items, from DMG 213, which just specifies that you "activate" an item. Once you activate the item, everything comes down to the wording of the item. Rings make for good examples of why this is complicated. If a ring can be activated, the benefit of wearing said ring is the ability to activate it, whether by command or use-activation (aside from any other continuous effects, etc). Here are some examples of activation wording:
Ring of Animal Friendship (DMG 230): On command, this ring affects an animal as if the wearer had cast charm animal. So it's as if the wearer cast the spell, but the ring affects the animal. There's no implicit range restriction on the ring other than the spell being cast by the wearer. So with the wearer having the ring on hand, this isn't a problem, but by strict wording it would seem that mimics the wearer, not the item, casting the spell.
Ring of Blinking (DMG 230): On command, this ring makes the wearer blink, as with the blink spell. Here there is no implicit range restriction, but the ring creates the effect as opposed to the wearer.
Ring of Invisibility (DMG 232): By activating this simple ring, the wearer can benefit from invisibility, as the spell. Again, no implicit range restriction.
... These are a few examples of such wordings, but you can look at the Ring of Friend Shield, Ring of the Ram, Ring of Three Wishes, Ring of Shooting Stars, and see that these are all command-activated with the ring casting the spell, not the wearer. So as a DM, I'd rule that, unless an item's wording specifically allows the wearer to cast a specific spell (I don't know of any), as opposed to the ring casting the spell or allowing the wearer to be affected by the spell, then the spell is cast by the item, which, given the above examples, seems to be the intention of the RAW. A custom item like a Ring of Fireball would have to have explicit wording of "this ring enables the wearer to cast fireball 3/day" or something like that, rather than "up to 3/day on command, this ring casts fireball as the spell". But as I mentioned, from the existing items in the DMG, it seems like items are meant to cast spells themselves upon activation, rather than directly enabling you to cast a spell (of course Ring of Wizardry is different, explicitly increasing your spells/day).
The intelligent items portion is more vague.
If you'd like the benefits via phylactery, you'd probably have to deal with the ego of the item, but the above applies.
Page 271 in the DMG says ego clashing occurs with a possessor, not necessarily a wielder/wearer. If you're going to transfer benefits via phylactery, I guess technically that'd make you a possessor, so I would imagine you'd be dealing with the ego of the item constantly.