Not necessarily......
I work a few days a week for the manufacturer of Cif/Jif. They went through a brand harmonisation process a few years back to reduce 1,400 brands to 400. That's why a lot of products seemingly disappeared from the shelves, whilst others (eg. Dove) expanded from being a bar of soap to an entire line of cosmetic products.
Decisions were taken on an individual case basis. Jif became Cif to fall in with the rest of the world. Sure is still Rexona everywhere but the UK. Lynx is Axe everywhere but UK and Ireland.
It's whatever made the most sense on a brand by brand basis.