Why don't we have allocated cabin baggage space just like seats?
Passengers do usually have some allocated space - under the seat in front of them. No-one else is going to claim that...

For the overhead lockers, a quick calculation (as @mins suggests) - seat pitch averages about 30", so there is going to be approximately a 30" slot above each row for "its" locker. On a 3-4-3 widebody, this would be ten seats sharing 4x30" of locker space, equivalent to a 12" slot each. On a 3-3 narrowbody, the six seats would share 2x30" of locker space, or 10" each. Not all seats have an accessible locker above them - eg emergency equipment storage - and there is some space taken up by the opening/closing mechanisms, so in practice it would be an inch or two less than this overall.

Meanwhile, almost all airlines give a maximum smallest dimension of their cabin bags - ie narrowest width they can fit into - of 20-25cm, or 8-10". So one standard-sized bag per person approximately fills the space given, assuming they can all stack neatly on their sides.

So, in theory, yes, you could try allocating space, one ~8-10" bag slot per person, and the overall space is available. But this would almost certainly mean putting in dividers to avoid disputes (you can imagine the disputes, right? "That man's bag is taking up some of my space!").

The dividers would mean that you couldn't put anything weirdly shaped in there - say, a walking stick - and would mean there would be problems with bags that don't quite fit the limits. At the moment, they can bulge a bit and be squashed in with everything else, so the passenger in seat 11A with a slightly outsize or awkwardly shaped bag can benefit from the fact that 12C's is quite slim, and everyone is happy, but once you have rigid dividers that becomes impossible. 12C's would rattle around, while 11A's doesn't go in, and overall the space is used less efficiently.