i've never done wdw solo but we've run into solo travelers quite a few times. on rides sometimes they've sat with us...test track and rnr for example, others like thunder mountain a gentleman was in the seat behind us alone. i think it just depends on the ride. i wouldn't think a ride like haunted mansion or buzz would have you ride with a stranger. when i was with a group of 3 we did tsm and they did one side with 2, the other single.
It really does depend. On the bigger, busier rides - mainly rollercoasters - you probably will get seated next to someone. I've never been paired up on Splash, though. It's mainly the coasters.
On the other rides, smaller dark rides, etc. you'll either get your own row, or at least your own "space". Sometimes, your own vehicle (I went on POTC completely alone last month - something I've done before - and it's creepy and fun).
Best guess is just go with the flow. I've been to all the parks (Uni and WDW) solo many times, and I can think of exactly twice I've felt weird. Once was on TT when a family seemed a little preplexed why they put me in the car (someone said, "Well I guess he's in our family now" - in an odd way, not especially welcoming, and I just smiled and didn't really say anything and went on the ride). The other thing was on Kali - they put me with a family in a raft, and they sat me next to the tiny wife and we had to share a seatbelt. That was just...odd. The husband and wife looked at each other for a sec, shrugged, and we just went for it.
In both cases (out of the 100's and 100's of times I've gone on rides solo), it wasn't that big of a deal even when it was a little weird. Most people just go with the flow.
(Although it is one of the times I wish I had a tattoo on my forehead that said, "Hey, I'm gay" - kind of like when I'm at a teenybopper concert - I always get sat on roller coasters with young girls and I want to be like - no, if I smile and say hello I am not hitting on you, haha.)