Can I download the OS X Mountain Lion update and share it with my family?

Yes, you can burn a DVD from it. The instructions are the same as for OS X Lion and need to be performed before you upgrade (otherwise you don't have the Install OS X Mountain Lion application anymore).

If you don't have the download anymore, you can probably get it from somewhere else; since you bought it this could be legal depending on your jurisdiction. The InstallESD.dmg MD5 checksum for Mountain Lion as released yesterday is 8b4869920cd740414fe6b7e3f0b1be3e.

When installing OS X ML from scratch, it (repeatedly) asks for your Apple ID, but entering yours can be skipped in all of these dialogs.

Note that the following restrictions apply (license agreement PDF, emphasis mine):

B. Mac App Store License. If you obtained a license for the Apple Software from the Mac App Store, then subject to the terms and conditions of this License and as permitted by the Mac App Store Usage Rules set forth in the App Store Terms and Conditions (http://www.apple.com/legal/itunes/ww/) (“Usage Rules”), you are granted a limited, non-transferable, non-exclusive license:

(i) to download, install, use and run for personal, non-commercial use, one (1) copy of the Apple Software directly on each Apple-branded computer running OS X Lion or OS X Snow Leopard (“Mac Computer”) that you own or control;

This is in contrast to the section A above it, restricting the license of a preinstalled Mountain Lion to the single computer it came bundled with. So in some situations, you are allowed to install one ML purchase on multiple computers.

So it could be legal to install it to other computers in your home you control (e.g. your children's), but it's prohibited to share it more widely.