The GMail account on the phone is used internally to identify the recipient of the C2DM message. First a client registers itself, then (when a C2DM message is sent) all registered clients receive the C2DM message. More than one client can register them self as recipent for a C2DM message.
From the Google C2DM site (see Registering):
Intent registrationIntent = new Intent("com.google.android.c2dm.intent.REGISTER");
registrationIntent.putExtra("app", PendingIntent.getBroadcast(this, 0, new Intent(), 0));
registrationIntent.putExtra("sender", emailOfSender);
startService(registrationIntent);
The parameter app identifies the package name used by the registration process aside with the sender ID (in the code above the var emailOfSender). You see that the GMail account on the device is not used in the code, but will be used internally to identify the mobile device on the C2DM server (either the Android implementation provides the client side GMail account directly or an ID linked to the GMail account).
Google says: It requires devices running Android 2.2 or higher that also have the Market application installed. This is because the Market app maintains the connection to the C2DM server. The registration ID is different for all devices. Prior for sending C2DM messages from your server the client has to tell the server the registration ID.
When you want to create an application for sending C2DM messages (on the server side), you need also an GMail account (the SenderID we used on the device). Typically the pattern "one GMail account per application" is used. When you register for C2DM you have to enter the SenderID and the namespace of the receiving Android app into the registration form - exactly the same information as used on the client to register the device.
Both GMail accounts are not public. The relationship is n-1-m, which means n clients and m server are registering them self at one C2DM server. Only Google (the C2DM server) knows which GMail accounts are used.
I have one real life GMail address which is used on my mobile. I have on debugging GMail address which I use on my emulator. Then I have 3 GMail accounts for every C2DM capable application I wrote.