Most of the time that I see someone with a company logo on their shirt I do not readily assume they work at that company.
Personally, most company swag I've worn once or twice and then throw in the Goodwill bag. So it's possible there's some random folks wearing a $NoNameCo shirt, but I'd wager they're just as likely recycled into something else.
I've gotten one single shirt from a hackathon I considered wearing based on their well designed logo, but the entire back is covered in sponsors rendering the entire thing garbage.
Beyond that, I prefer to order t-shirts with my own design; $random-company swag frequently ends up as home/nightwear for either me or my SO.
I couldn't really give a shit what I look like most of the time, however, so your mileage may vary. But I don't have to buy t-shirts...
I'm shocked that, as an employee, there's no "company store" where I can buy an extra $Company coffee mug, t-shirt, polo shirt, etc.
Often times the internal schwag is made by name brand outfits. The stuff they give out can be lowest bidder stuff. Also designs can be somewhat different for the internal vs external versions.
That said, perh for many smaller cos both are one in the same.