Former tire buster for BF Goodrich here... this is the oldest tire con in the book, dating back to when "radial" tires started to come on the market (just shortly after the invention of the wheel and the discovery of fire). We routinely told customers that you could not replace just 1 tire because you can't run radials with non-radials. And this was the truth. However, it quickly became clear that we could also use this as an upsell technique: "You can't run 2 radials with 2 non-radials... you can't run 2 different radials on the same side of the car... you can't run 1 radial from 1 brand with radials from a different brand... and --- last but not least --- you have to replace all 4 tires if you're going to replace just 1."
Sold a lot of tires that way. There's only a couple real issues with replacing tires.
1) Don't mix sizes
2) Rotate tires front to back, not side to side. You don't want to run tires "backwards".
3) If you replace 1 tire try to replace it with the identical brand and size; having mis-matched brands can mean slightly mis-matched sizes, even if the sizes are the same.
4) Otherwise, if you replace 1 tire the remaining tires should be reasonably close to new. If they're not, then replace the other tire on the same end of the car (ie., replace both front or both rear tires). This is just to help with steering control, especially on the front. Having 2 tires with dramatically different wear on the same end of the car can make handling funky, and will wear out the new tire much faster.
Otherwise... it's good to know that the old "YOU'LL KILL YOURSELF IF YOU DO THAT!!!" sales pitch is still alive and well.
