^ bl has it right. The fingerings for the written notes remain the same for different instruments, even though the actual pitch is different. This makes it so that saxophone players can switch between an alto sax (in Eb) to a tenor or baritone sax (in Bb), for example, without having to overly think about where to place their fingers.
If I got this right, the fingerings stem from a recorder, which is in concert tuning. Playing for example a G on a recorder uses the exact same fingering as a G on a tenor sax (which would sound like an A).
Almost right. All modern woodwinds use the Boehm System of fingering, named after
Theobald Boehm. But Boehm was a flautist. Technically a recorder is a very simple member of the flute family, but the fingering system started with the concert flute.
People ask me a lot "how many instruments do you play?" and hate it when I tell them "It depends on how you count them". I play saxophone, which means I can play alto, tenor, soprano, baritone, or even bass saxophone, because the same fingering produces the same notes. So does that count as one instrument or five?
D on a flute is the same as a D on a saxophone, an E is an E, etc. And that works for regular (concert) flute, alto flute, bass flute, and piccolo. Is that one instrument or four? They're all played the same way. There are a couple of notes with quirky fingerings, but for the most part, if you can play one woodwind, you can play them all.
It has to be this way. Since a tenor sax is larger than an alto, it plays in a lower pitch range. But you can't worry about which instrument you're playing and use an entirely different set of finger combinations for each instrument. Instead, the notes are all the same, and you transpose in your head or transpose the written music itself.
The one exception is the clarinet family. For some reason known only to the inventor of the clarinet and probably Satan himself, when you switch registers on a clarinet, you go down a 12th, not an octave, so for half of the instrument you really do have to "think" in a different range. This is why clarinet players are all psychotic.