... equipment worn or carried ... is similarly enlarged by the spell.
The spell causes you to go up one size category. Since there aren't pre-existing rules for a character changing size category, it then goes on to mention how to apply that to a character - doubling height and multiplying weight by eight. Then at the end of the spell is that note I quoted above: similarly enlarged in this case must mean simply that the equipment is increased one size category, not necessarily that its weight increases by a factor of eight. Looking about the Core Rulebook, there are pre-existing rules for increasing a weapon or a suit of armor by a size category, and they explicitly state that doing so doubles the weight of the equipment in question.
A strong implication, though not a solid "rule-as-written", is that creatures' weight capacities increase and decrease along the exact same scale, as you mentioned, doubling or halving as you go up and down categories. Thus a creature with a specific Strength and Dexterity can use and carry the same equipment as another creature with the same Strength and Dexterity, regardless of their respective sizes, so long as their equipment is also correctly sized.
Finally, equipment refers to items that can be equipped. Most non-weapon, non-armor items in the CRB do not have size-scaling rules because they are assumed not to scale to sizes. To pick a random example, the bathtub has a fixed cost, fixed weight, and permits a Medium creature to bathe in it specifically. While a GM could extrapolate a Large bathtub (and it would probably weigh twice as much), by RAW they don't exist - and, since they're not equipment, the enlarge person spell would not create one, even if a character were carrying a bathtub at the time they were enlarged. Typically, only items that can be wielded (not simply held) or are worn or otherwise attached to the body are equipment. Storages like backpacks and belt pouches should probably be considered equipment, since they're worn, but things in them like alchemist's fires, potions of cure light wounds, firewood, and waterskins should not. Multiplying an alchemist fire's weight by eight wouldn't create a Large alchemist's fire - it would actually BE eight alchemist's fires in one big bottle. (Exploiting that loophole, in the case that a GM did multiply gear's weight by eight, would be difficult for a PC to do safely, since enlarge person ceases to affect items fired or thrown or otherwise removed from the enlarged person's, er, person.)
"Specific trumps general" probably doesn't apply to that line of enlarge person for the simple fact that similar doesn't mean identical and the wording was likely chosen to allow the spell to function as a "buff" without having to spell out yet another time in the same book that increasing equipment size is x2 weight, not x8 (rather than a "debuff" that immobilizes a creature under the weight of its own backpack). If the spell actually applied the exact same changes to the equipment that it did to the character, it would have to increase the items' Strength ratings, and decrease their Dexterity and AC ratings. Most of those operations are impossible (or else would result in paralyzed items with 2 Str), although your armor and shield would suffer an additional -1 to AC in addition to your new -1 size modifier, so a Wizard's AC might only decrease by two when enlarge person is cast upon them, but a "sword-and-board" Fighter would lose a total of four AC. Likewise, composite bows are crafted with Strength ratings... as your Strength increases from 18 (+4) to 20 (+5), your bow's Strength rating would decrease from +4 to +2, right?
**TL;DR: Similarly increased refers to the size increase, not the weight increase, and doesn't replace the existing rules for increasing the size of equipment. Non-equipment items don't change sizes. In most cases, you can carry more while enlarged, not less.