No, HP and generic "brown sauce" are very different animals. TBH I don't much like either, preferring ketchup, but my wife will only touch HP. She won't touch no-name generic brown sauce at all.
Oh I know, and I've traveled so I've had brown sauce before. But condiment preferences is all about what you're used to, so I prefer ketchup. Just find it amusing how adamant British people are about the superiority of brown sauce.