Do two handed weapons add 1.5 x the strength modifyer or is something else done for extra damage?
No, they don't. But one should take the different damage dice into consideration.
a non-magical one handed (martial) weapon uses a d8 (or d6 if they have funky traits). Two handed weapons use a d12 (d10 if funky).
Average rolls:
d12->6.5(2H)
d8->4.5 (1H)
You can see, that a two handed weapon does two points better, on average, than a similar one handed one. (Though that was also the case with PF1-weapons, more or less)
But now consider that a magical weapon uses multiple damage dice.
So on a +3 two handed weapon you use 4 damage dice, each doing ~2 points more. The difference becomes greater on higher levels.
I think this is a quite elegant way to give two handers more punch. In PF1 we had a lot of flat +x, which were boring.
For example a paladin (~level 10) using smite + power attack adds +10 for smite and +9 for powerattack +9 for STR (two handed +6 x1.5) +3 Magic Weapon. (All estimates- I didn't check the rules)
In that case it doesn't even matter what you rolled on that one d12 roll with all these flat bonuses...