I recently had a saxophone student perform a repertoire piece with some altissimo technique in it, and a non-woodwind-playing musician asked me afterward about the instrument’s extended range. This led to further questions about “altissimo” on other instruments. The answers are a little complicated, but here is some information:
The term “altissimo” suggests an extreme high register. The term is widely used by clarinetists and saxophonists, with essentially the same definition: pitches in the instrument’s third register or higher. Basically, this is notes above (written) F-sharp-6 for saxophones or (written) C6 for clarinets. (It’s not really that simple if you factor in alternate fingerings: a clarinetist, for example, might use a trill fingering to produce a D6 in the second register, or a saxophonist might use a “front” fingering to produce E6 or F6 in the third register.) For clarinetists, using some of the altissimo register is a pretty basic technique, part of the instrument’s “standard” range (which extends maybe to G6, depending on who you ask), and accessible to, say, an intermediate-level high school student. For saxophonists, altissimo is viewed as a more advanced technique, outside the “standard” range, perhaps accessible to college-level musicians or motivated high schoolers.
The word “altissimo” isn’t used much in the flute and double reed worlds, though those instruments’ third registers are widely used even by intermediate-level players. (Christopher Redgate does use the term in his writings about oboe extended techniques, but arbitrarily defines it as beginning at G6, well into the instrument’s third register.) The flute’s third register begins (basically) at D6, the oboe’s at C-sharp-6, and the bassoon’s at E-flat-4.
In short, all of the woodwinds do have an “altissimo” range in the sense that they have a third register and higher. But not all of them use that terminology, and those that do differ on whether the altissimo range is “standard” or an extended technique.