I noticed some weapons, like the Elven Thinblade, can qualify for Weapon Finesse despite being classified as one-handed weapons. Does this mean I can wield it two-handed for 150% of my Dex bonus? Alternately, how can a Swashbuckler/Rogue multiclass apply Dex to attack rolls?
They are not mutually exclusive
And in fact there's even one weapon that's two-handed but finessable (the Spiked Chain). You can Power Attack with a finesse'd weapon as well, unless it's classified as a Light Weapon. Did the designers intend this? We'll never know, but the RAW is fairly clear on the matter.
There's no text in Weapon Finesse or the section on using a one-handed weapon in two hands that forbids it.
Nothing in their respective descriptions states that they are mutually exclusive (Though Light weapons state they can't be used two-handed, and Natural Weapons cannot be two-handed for obvious reasons)
If the weapon says it qualifies for Weapon Finesse, it means just that: You can use Weapon Finesse with that weapon. It does not imply anything else on its own.
Therefore: No, you cannot use 1.5*DEX to attack with a two-handed Finesse weapon (nor can you use 1.5*STR with any 2H weapon; it's for damage.)
Swashbuckler/Rogue can use DEX for Attack by using a weapon that can be used with Weapon Finesse (that Swashbucklers get at 1st level). Such weapons include: all light and natural weapons, Rapier, Spiked Chain, Whip, and others not in the PHB.