The Ordnance Survey TOIDS (Topological ID) are indeed unique IDs for OS features. The reason there are sometimes 13 and sometimes 16 digits after the 'osgb' bit is because they changed the specification but some data had already been released with the early 13 digit TOIDS so they were stuck with it.
Many systems designed to work with OS data prepend the leading zeros for consistence and ease of parsing later, but they are not part of the original spec and can be deleted with impunity. In fact, OS' own TOID validation service requests that you omit the leading zeros.
The 'osgb' part is common to all OS TOIDs and really just identifies a TOID as an ID belonging to and/or originating from OS data to enable a quick distinction between data from other sources. If you can guarantee that your databases will never acquire data from elsewhere it might be safe enough to lose the 'osgb'.
However, having said all that, the TOID is the basis for receiving change-only updates from OS, so you would have to reverse engineer all your space saving for each update. Also, having headed up a team that was developing automated processing methods working on the entire OS MasterMap dataset plus data derived from other sources, I can say from experience that having the full TOID complete with 'osgb' was very useful for tracking the provenance of a problematic features in some combined and derived datasets when you are a few steps from the 'raw' data. Given that disk-space is cheap these days, I'd rather buy a slightly larger drive than save the space if required. Even if you keep your 'raw' data pure (recommended!), GIS data gets used and abused, so having an idea of the provenance after OS data has been combined or analysed with other non-OS data is very useful. Knowing the full TOID in such circumstances also means you know whether you have to re-compute your derived data following an update from OS. Without the full TOID, you cannot guarantee you know where a feature came from after processing.
In addition to the above, the OS data also contains a number of attributes of the form 'ReferenceTo'. These are references between data-sets that provide additional information on how they interrelate (e.g. how lines might relate to polygons - i.e. a road casing line relates to the matching road topological-area polygon). Any stripping exercise you perform on the TOID itself must be matched on these fields as well and that increases the risk of error during your stripping process.