If you can boot to it, you can probably update the chipset and other drivers for the new hardware. You might not be able to boot though. Some backup software have a "redeploy" feature to restore a backup to new hardware. I managed to boot up a new disk to a new motherboard and CPU once and fix it stable. It may be hit or miss though.
Copies of non OEM windows 7 can be used on up to 2 CPU's or motherboards, so for an Identical replacement given you bought the non OEM version of windows 7 and were on your first round of hardware you would be OK driver wise and wouldn't get the windows 7 not genuine messages. It is possible to just migrate the drive to a new system but you would need to uninstall all your current drivers and make a few changes to the registry in advance, the process would be more complicated than backing up data and reinstalling. The best way to speed the process up is to use an ssd or second hdd for a boot drive, and a separate drive for storage, all your installers can be kept on the storage drive to make it even easier in case you change hardware or have problems with your installation.