When I play with a band, on whatever instrument, I listen to what everyone else is playing, and do whatever I feel is "right" for the music at that time. Sometimes that's holding down a simple bassline. Sometimes It's playing 5 or 6 note chords and using 5 effects pedals. Sometimes it's playing high up the neck, playing melodies, bending notes etc. You might call that a "bass solo". When I play on drums there are some sections where I just play mad fills. You might call it a "drum solo" but it fits in the groove, and is right for that part of the song.
It's not showing off!
I guess most posters here are talking about a rock band where the whole band drops out, or the singer introduces the players, and then someone fret-w***s for a couple of minutes. I'd agree that there's nothing cool about that
On the point of "suitability" - I get the historical point but double bass works well in jazz as it can generally cut through. As long as the playing isn't too fast it sounds cool. Amplified electric bass is just as suitable to soloing as any other instrument