In Fallout Shelter, the first thing you can do is pick the three-digit number of your vault. This seems significant, but I suspect it is not.
Other than labeling your vault and helping you distinguish between playing multiple vaults simultaneously on the same device, is this number significant? Does it set a standard randomization seed or starting layout, or are there specific easter eggs with a given vault number?
Primarily, the vault number is just a means to identify a specific vault, so that players can easily identify a given vault between other options.
If you consider Fallout Shelter canon, each vault number coincides with an actual vault, in the Fallout universe. In the canon, each vault played host to a specific experiment. Experiments varied, from having a vault with deliberately-dim lights, to having a vault that contaminated its residents with excess radiation, to having a vault comprised entirely of children.
Players wishing to engage in a deeper level of immersion may wish to select a number associated with a known vault, and recreate the experiment. However, there is no game play mechanic that makes obvious game changes based off the designated vault number, and not all experiments can be recreated with the available game mechanics.