Steam – this message when somebody wants to send gift money with the item

The actual sign-in process returns very little information to the third party service/website - nothing more than your Steam Name, and your OpenID URL (which contains your 64-bit Steam ID).

Steam OpenID Provider
Steam can act as an OpenID provider. This allows your application to authenticate a user's SteamID without requiring them to enter their Steam username or password on your site (which would be a violation of the API Terms of Use.) Just download an OpenID library for your language and platform of choice and use http://steamcommunity.com/openid as the provider. The returned Claimed ID will contain the user's 64-bit SteamID. The Claimed ID format is: http://steamcommunity.com/openid/id/\

Any further information that is returned is made by further calls to the Steam Web API (as shown in the answer by @longtgomjr), and can contain any manner of public information about your profile.
It's worth noting that most data isn't available unless your profile is public. This is not a part of the login process, and the retrieval of this data can happen regardless of you logging in.

When you are given a game as a gift on Steam, it will land in your inventory, but you won't be able to play it right away. To play it, you need to add it to your library. This allows you to re-gift a game, in case you've received one you don't want and would like to trade it for some other game.

Although I have never received two or more copies of the same game as a gift, I believe that you can receive as many copies of one game, but can only add one of them to your library. In fact, if you buy a bundle which contains a game you already own, a copy of that game will land in your inventory as a gift. There is no reason to believe you can't hold multiple copies of the same game in your inventory.