I know about true or false harmonics tones on various instruments but I do not understand what an altissimo is. Just very high notes? Can you produce them on every kind of saxophone? Can you play a melody with them?
Altissimo is just a name for notes above what is (or used to be, at least) considered higher than the "normal" range of the instrument. Most saxophones can play up to written F or F-sharp above the treble staff normally just by using side keys that open tone holes farther up the saxophone to make the air column shorter along with the octave vent opening. The notes higher than that are considered "altissimo." Altissimo can be played on any saxophone (of sufficient quality), and it's still called altissimo regardless of whether it is being played on alto, soprano, tenor, etc. Generally speaking, the lower saxophones have more altissimo notes available than the higher ones, and it is easier to play altissimo on lower saxophones than higher ones. The fingerings for altissimo notes are not the same on all saxophones.
Overtones can be, but are not necessarily, in the altissimo register. With the fingering for written B-flat just below the treble staff, the lowest note on most saxophones (except baritone, which usually has the A a semitone lower on professional models), it is possible to manipulate the tongue (mostly) and the embouchure (slightly if at all) to produce notes in the harmonic series above the fundamental written B-flat. The same can be done with other "low note" fingerings, though as the fundamental pitch increases, the difficulty of producing the overtones increases, and the intervals comprising the overtones become somewhat less "correct" due to the effects of the open tone holes.
Altissimo is now expected of all professional saxophonists, and many works for advanced saxophonists use altissimo just as they would use any other notes in a melody. Overtones as I described above are rarely specified in a piece, though some composers use them as a "special effect" occasionally. Saxophonists tend to practice overtones as an exercise in learning to control the tongue position within the oral cavity, which can help in producing notes in the altissimo range.