An important factor in improvising fluently (such as in a jazz context) is a collection of vocabulary. Broadly defined, this could include rote-memorized “licks” plus all kinds of other material: scale- or arpeggio-oriented patterns, for example. With improvisation, you don’t have the luxury of practicing your solo note-for-note; instead you have to develop a large pool of available material, and learn it so well that you can mix and match it on the fly.
If you double on multiple instruments, the vocabulary pool isn’t really portable. You can bring your improvisational ideas with you from instrument to instrument, but you won’t be able to execute them smoothly unless you have put in the practice hours on each instrument separately. If you are doubling, say, saxophone and flute, you might find that the fingerings are similar enough that you can make a few things work, but it’s too easy to paint yourself into a corner, or to catch yourself using “close enough” fingerings that really aren’t, or to play with unsatisfactory tone or intonation. To do it right, and have the colors of multiple instrumental voices available to you as an improviser, treat each instrument like it’s your only one.