Does buying a new mac also get you Lion on your Apple ID?
No. The copy of Lion you get with a new Mac is only available via the recovery partition on the hard drive and this version of the OS is tied to the hardware.

From this MacWorld article that talks about how the version of iLife you get with a new Mac is a full Mac App Store version you can install on multiple Macs in your house:


  When you reinstall Lion via Lion Recovery, it seems Apple uses information about your Mac's hardware to verify that it has a license for Lion.


Or as this summary of the MacWorld article puts it:


  Unfortunately, as Macworld points out, this trick doesn’t apply to OS X Lion itself — there’s no DVD or thumb drive included with a new Mac, which means the big cat only lives on your new Mac’s recovery partition, which appears to be tied to the hardware itself — meaning you can’t install it on your other Macs without ponying up the $29.99 from the Mac App Store.


Would be nice if you got it for your other Macs for free when you bought a new Mac, but for now it seems like it'll still cost you another $30 to bring all the Macs you own up to the current OS rev.