I think the title says it all. Could you sign a dual voltage unit in only DC mode, for example? Or a hybrid locomotive in only under-the-wire mode?
Or if you sign traction is the idea that you can "do everything"?
Yes it certainly happens. There are many Drivers who sign dual voltage stock but only sign routes of one voltage.
SE Drivers sign 375/6 and 377/5 in DC only.
SN Drivers (except for a very small amount at Norwood depot who sign the Clapham to Watford route) sign 377/2 and 377/7 and 387/2 in DC only.
LO Drivers (at New Cross Gate depot) sign 378/2 in DC only.
LO Drivers (at Chingford depot) sign 710/2 and 710/3 in AC only.
GN Drivers (except for a small amount who sign the TL route too) sign 387/1 and 387/2 and 387/3 in AC only.
GWR and HX Drivers sign 387/1 in AC only.
LNWR and WMR Drivers sign 350/1 in AC only.
Eurostar Drivers sign 373 in AC only.
So it definitely happens. It just means they are only trained on the AC or DC part (as applicable) as training on the other mode is unnecessary.