I've ridden with groups small and large, with individuals and on my own. Each have their own place but I must admit to greatly enjoying the singular experience, even if (as the op remarks) you'd sometimes want for company to share that experience with. It kinda validates it, makes it more real. I remember once standing on a bridge in the small hours, all shrouded in mist, looking at the river thinking that life really didn't get much better. A singular and life affirming experience. Nothing 'lonely' about it.
If you're pootling about locally, using your bike as everyday transport, bobbing out to the local shop etc then you will go alone. (How does that phone conversation go - "I'm just cycling down to the co-op to get some rich tea biscuits, fancy coming?")
I've tons of admiration for people that use their bikes like that, for everyday needs and purpose. No need for all the gadgets and Lycra, just a pannier and pump and off you go. I reckon there needs be more recognition of that kind of cycling.
There's an old boy near me, must be in his 80s, and I'll see him out doing his circuit, visiting a friend nearby, pushing his bike up a couple of hills, but simply using the bike because he wants/needs to go somewhere. Top stuff matey, top stuff.