In the UK, we give names to "lines" or routes rather than the train. The definite article is usually used in normal speech, for example "The Victoria Line". A search for "Victoria Line" reveals that this is almost universal in most kinds of texts such as news reports, discussions by or about passengers, services, railway administration.
In timetables and also their announcements of engineering works, delays, etc, the train companies seem to avoid the definite article by using a written style that is less discursive and which uses formatting that favours brief titles (displaying the name of a line simply as "Victoria Line", for example). Extensive examples can be compared on the Transport for London website.
The only named train I can think of offhand is Eurostar. This could, in normal speech, refer to either the services or to the physical trains that it uses and which are highly distinctive to it. We talk of "taking the Eurostar" but also of "going by Eurostar". British people wouldn't assume this referred specifically to either the physical trains or to the institution and context would be the guide. Nevertheless, as seen on the Eurostar website, the definite article is largely avoided by the operators and the name is used in the manner of a brand name.