If PC can have free online, why can't consoles?

You don't pay for online on PS4 but for the "free" games, of course.

So can I opt out of the free games but still get online?

Of course (if you have a PS3).

Can I get the free games without paying for online?

Sure. There are a lot of free2play games.

PC has free online mostly because there's too much competiton on PC for that to happen. If Steam started charging for Online people would just buy games somewhere else. The money Steam would lose from sales would outweigh what they would make from subscriptions. Consoles are closed systems that don't have that problem. If you don't want to play for Online on your Xbox One, but still want to play multiplayer, what are you going to do? Dump your prefered gaming platform over $60 a year?

On top of that, I think Steam takes a larger cut from games than Sony/MS do. I could be wrong but I think it's something like 30% vs. 20%.

So it's essentially console manufacturers charging because they can?..

They realized they can so they did. Not that it would not be justified to make consumers pay for infrastructure costs, though.

Partly, but it's also becuase they fund themselves differently then someone like Valve does with Steam.

Sticking with the Steam example, they basically make publishers/developers pay for online through a higher cut of the total revenue wheras Sony/MS charge the people using online multiplayer directly for it's use.

Mind you, if either Sony or Microsoft had data showing that they could make up the lost revenue through increased sales by making online free, they would. But it doesn't hurt them to charge for it, which means it wouldn't really help them to make it free.