Is it possible to get a new Minecraft *game* id without getting a new *user id* / buying a new copy of Minecraft?

That would be very bad, if it would be working. It is of course not working. The game is linked with you account and it is not possible to change that.

The ID was earlier your in game name. With your name you will be identified from the servers. The name is not changeable in an easy way. The only way is to contact the support (but they wont change it).

Now it is working with a unique ID (v1.7.6). Means you are able to change your name but not your ID.

This version provides complete support for identifying players through unique IDs and not by names, so that no matter what a player is called they will remain banned/whitelisted/oped/etc on servers.

_{from mojang}

The reason why you cannot change your ID is because server owner have a problem to identify you. Means if you are banned from a server, you should be banned and not have a chance to join.

Answer for your edited question:

The functionality on a server is basically the same as on a local game. You actually can not have more "games", but you can have more worlds! On a Server it is exactly the same (if the owner has build it like so). If the server owner has nothing edited and just has a bunch of different maps, you have a "new game/status" on every map. There is nothing limited.

Anyway the chance that there is a server that is build up like that is very low, because everyone wants to keep his items and build just one house and that on just one map. If there are more than one map it is likely a mini game map, pvp, skyland or whatever.

There is no possibility (legal and playable on all "online" servers) in any way to get a new status/"game" on a server by changing anything at your client or on your account.

The only way is to ask the server owner to delete your user data from the server.

To directly answer your question: No it's currently not possible because Mojang did not program it that way.

Your question is essentially: Why did the developers of a game not implement this feature? Who knows?

I agree with you it would be a great feature to on each server be able to create progress saves as a different entity in each save but still be identifiable as the same person but it is simply not a feature at the moment.

Yes, there are multiple name changing mods out there but a bunch of them are either within hacking clients or really glitchy. Also, in order for the name change to actually work, the server must be in offline mode. There is no easy way to find out if it is actually offline.

The hacking client i use is called nodus in case you were wondering. Inside the nodus hacking client, in order to change your name, just enter either the multiplayer menu or the options menu, there should be an add account button. Press that and fill out the desired boxes.

Note: They are just called hacking clients, they don't do any literal hacking, they just allow you to break blocks faster, add a forcefield, walk on water, fly, use the x-ray mod, and much more BUT most servers also have anti hacking client addons or mods that will just ban automagically you from the server if you are caught using them.

Another note, if you wish to start fresh on a server then politely ask the owner of the server to delete your account by going into the the worlds folder and entering the players folder and finding your minecraft id. The path of this would be [server_folder]/[world_name]/players. What ever is in the [] is the whatever the folder might be named.