If your newer ASA 5510 has a more recent version than the (very) old 6.2 that's on your 5505 you can copy off the asdmxxx.bin file and use it on both appliances.
You mixed 6.x and 9.x nomenclature in your original posting. ASDM's latest release is 7.12(1) as of today and the latest release for the ASA 5510 is 7.8(1)150.
9.x would be an ASA software release.
Note that you can download a free ASAv (virtual machine image) and use in in your home lab. It requires a cisco.com account but does not require a support contract or partner affiliation. The only limitation of the free (unlicensed) edition is that it will only allow 100 kbps throughput. It will run fine on the free VMware ESXi hypervisor.
@Marvin Rhoads: I believe you do require an active contract with Cisco - not just a login. At least that is the message that keeps coming up when I try to download it from the link provided … and yes I am logged in.