The underlying idea of bitcoin is that there is no central body that is responsible for the money. Therefore there is no official organization that owns the name bitcoin. This would also not be possible (anymore) because bitcoin has now become a generic name: a usual name for a type of currency. So, an application of Bitcoin for financial or administrative services would resolutely be refused by, for example, the EUIPO as we have seen multiple times. Alternative trademarks such as Bitcoin Analytix are accepted on the other hand. However, such trademarks give little protection to the word bitcoin itself.
The name bitcoin can of course be registered for completely different products and services. Products and services that have nothing to do with a currency. This is often done, bitcoin is a popular name. We see applications for clothing and cosmetics for example. The coolest application is of Katja Fassin and concerns an application of bitcoin for candy. Bitcoin the new cryptocandy?