Looking at branding one thing we don't see anymore is rail companies using the name of their parent company in their operating name. When privatisation happened Virgin Rail Group, First Group, Connex and Arriva all added their company name to the branding i.e. Arriva Trains Northern. While Stagecoach and GB Railways would have logos that were similar throughout their operations. National Express (at first), Go-ahead group, and John Laing on the other hand didn't seem to have any similarities in branding across their franchises.
Now though other than the "by Arriva" I don't think there is much mention of parent companies being mentioned in logos, company names or typefaces.
I suppose it's for the best as many times people don't keep up with changing franchises. Also back in 2012 when the Virgin Trains contract saga happened I remember many non-railway people being worried that the Pendolinos were going to be withdrawn and replaced by HSTs and Pacers as that is what First Great Western used at the time. Silly I know but many people didn't understand that the parent isn't really an important factor in choosing rolling stock for a new franchise.