128Mbit chips begin to become commonplace
The engineers at Intel knew higher density chips would be available in the future, in either a x8 or x16 configuration, so they over-engineered their new chipsets to be capable of taking a 128Mbit chip. The 440BX chipset was old enough that they didn't know how the architecture of the chips or the memory controller would look, so the BIOS that was developed wasn't capable of supporting 128Mbit chips. When the 810 came out the BIOS was ready to be written to accommodate a 16x8 chip even though the chips themselves weren't yet available. Newer BIOSs also became available for the 440BX chipset, so once the new BIOS was loaded into that chipset it would also accept 16x8 build memory upgrades.

Manufacturers using the 440BX and 810 in their machines
The other confusing part about compatibility with these chipsets is when a computer manufacturer such and Dell or Compaq puts them in their machines. The manufacturer will write their own BIOS based on marketing requirements and depending on how they want it to perform. They will generally write a BIOS that restricts a machines abilities, so they can be efficient and have the same components inside different boxes and sell them with different properties. Later, as the new computers they sell need more capabilities, they write a new BIOS that will accommodate the higher density chips. In many (though not all) cases these newer BIOS's can be loaded onto a 440BX or 810 chipset and they will then accept the 256MB memory upgrade module built with 16x8 chips. Another method is to source the Intel BIOS that was designed or the 810 and flash the machine with that. No single solution to the problem is foolproof in every case, but the vast majority of machines can be updated this way.