How exactly does operating system update work on Android devices?
Most vendors/manufacturers modify the Android source code to ensure full compatibility with the devices hardware - if you were to update to a clean version of Android, it just wouldn't work properly (and it it even booted, most things wouldn't work at all).

Some manufacturers implement an OTA update feature - whereas some don't. Those that don't sometimes make you connect your phone/tablet to the computer and use their 'helper' program to do the update.

Even the devices that do connect to the internet and download the update are vendor specific - device specific even, it is just that the manufacturer let you do it that way.

As to your final question: Yes, through a custom ROM. Custom ROM's are versions of Android that have been modified by the Android community, based on the stock Android code, to work on your device. You should search google, or look at forums such as XDA-Developers for more info about ROM's, or to look for a ROM for your device.

Remember: All the hardware in phones are different. Only the manufacturer knows how to code for it, so that is why default Android code won't work with it - it would need propriety drivers.