Sure, if buy a 'national' ticket, which allows you to get on the next train by any operator going your way, then you can end up standing. As far as I am aware, such tickets have never been available on airlines!
If on the other hand you buy something more like a plane ticket, then a seat reservation is always included. And if you're buying more than one, then they will, if possible, be adjacent.
And as for 'you have a choice', many FR destinations from second tier airports are served only by FR. Excluding the major holiday destinations, the only ones from EDI which are served by other operators are CPH, STN, and Krakow, I think.