What troubles me more about the sequence is the fact that even after 15 years and a major refit, they hadn't changed the codes.
We need not assume that the string of numbers was a code, though. Not in the sense of being a secret password or a way to positively identify the speaker; those should be redundant with voice recognition already.
Rather, the numbers could simply be shorthand for what the person wants specifically. The "Code zero zero zero, destruct zero" bit probably serves that function, but all the "personal" strings of numbers are also first identified as being requests for a specific "destruct sequence". There might be plenty of nuances to it: say, Kirk first specifies that he wants a sequence that needs two more verifiers; Scotty concurs; and Chekov concurs. Kirk might also have chosen a destruct sequence that needs only his say-so, or a sequence that calls for the consensus of five officers, or a sequence that needs to get approved by the computer based on set criteria.
The string of numbers could also specify what the position of the person aboard the ship currently is (so it stays unchanged for Kirk and Scotty, but follows the Science Officer post when that transitions from Spock to Chekov). The CO position is unambiguous, but the ship might have plenty of Science Officers, and "1b2b3" might be needed to tell that Chekov is using the authority of the Chief Science Officer rather than the Assistant Astrobiologist.