I used to work with a guy who was very insistent that he was Czechoslovak.
Not Czech nor Slovak but Czechoslovak !
Seemingly his Mother was one and his Father the other and he took great pride in his hybrid identity and allegiance to a country which no longer exists.
Probably loads of folk like that in the former Yugoslavia and USSR as well.
A pair of diplexers (specific to the frequency ranges in question) would do the trick.
But depending on what reception is like in your area it may be the case that your FM aerial fortuitously picks up sufficient DAB signal for a separate DAB aerial to prove unnecessary ?
In Ireland its still theoretically an option (as in there is an end-of-school exam one can take in the subject) however its only available in a tiny handful of schools nowadays.
It was more widespread in the first half of the last century.
People do view TV Channels across timezone boundaries right now.
A lot of folk in the Netherlands and Belgium watch British TV and North America, Australia, Russia etc must have lots of TV channels with audiences spread across more than one timezone.
To say noting of migrant workers/families in various countries who largely shun the local TV channels in favour of satellite/internet TV from their home country. (Indeed the advent of internet TV is making the concept of TV schedules pretty obsolete anyhows)