It may be carbide but it is not necessarily the same type and grade as used on lathe tools; probably is not, routing wood imposes less shock loading but is more abrasive so a more brittle but abrasion resistant grade may be used.
But they can cut metal; we use corner radius carbide router bits up to 1/2" radius for putting a radius on aluminum parts in the mill. But this is at speeds completely unattainable in a little lathe, 1300fpm, and flooded with coolant. Our experience is that they cut best going as fast as possible and last a very long time on aluminum.
Regarding using a straight cutter for making a hole from the tailstock; probably not feasible.
This is because the tailstock has almost no sideways rigidity. When it is holding a drill there is really no side load. The drill follows its own hole, if you start it on center it pretty well stays on center. Reamers follow the drilled hole, again no side loads. Also both these tools have minimal or no clearance immediately behind the cutting edge; that is why they follow their hole they are guided by rubbing here.
A straight flute router bit in the tail stock will almost certainly dig in unevenly and chatter around horrendously making a very rough and oversize hole. These cutters do have clearance behind the cutting edge so if they are not held very rigidly they will dig in.
Possibly if a holder for the toolpost was made to hold the router cutter perfectly on center and very rigidly they might work. But then you might just as well bore the hole to size.