It’s the same end issue if you get shifted due to some bad neighbour chewing up resources (or your own account becoming large enough to being needed to be moved away in order to appease your own neighbours).
dreamhosters.com alias subbies point to the same box (they sometimes appear to be running on a different Apache instance – but there are several Apache instances running on each Shared host).
If you have multiple subdomains and aliases on a single account you’ve probably noticed that your apache instances can differ, as can the apparent IP, but they’re all on the same box.
If your primary account is moved then all of your endpoints change because everything has to be remapped internally to the new box and subsequently new IPs/Apache instances running on that box.
I don’t have my DH notes at hand right now to copy/pasta a quick query via hosted php or perl to echo which Apache instance any particular (sub)domain is attached to on creation, but if you log in over SFTP or shell you can take a gander at
/etc/ and it will show you how many instances are available on your box.
On my shared (
craig.dreamhost.com) there are over 30.