Why would BBC register TARDIS a trademark? Isn't it an image of the police force?

no it's a commercial icon for the Dr Who program and as such earns BBC a lot of money. It is a copy of an old ( obsolete) police box last used around 1960 ( ish). in the program it functions as a time machine an din early days of BBC showing the program there were still these boxes around. it proved to be a bit hit with crossing a high technology time Machine with an old fashioned bow that few would have an interest in.