On Steam, you have a Library and an Inventory. Your Library is the list of games you own and can play, but can't trade; your Inventory is the list of games you can trade but can't play.
If you want to play a game in your Inventory, you can add it to your Library, but games can't move in the other direction - once a game is in your Library, it's there forever and can never be traded.
You can view your inventory by going to Steam --> (Your Name) --> Inventory.
To add a game from your inventory to your library, click on it, then choose "Add to my Game Library" on the right.
..and I opened [the gift] it to see what was in it.
There is no such concept as "opening" a Steam gift. When you receive a gift, Steam tells you immediately what game you were sent, then gives you three options:
- Add game to my Library. (This lets you play the game, but not trade it)
Add game to my Inventory. Steam removed this option in 2019
- Decline (Returns the game to the gifter)
Can I have an opened copy of game X, but still receive additional copies of game X
Assuming you mean "If I have a game in my Library, can I also purchase/receive a second copy and put it in my Inventory?" the answer is yes, there are no restrictions on the games you can have in your Inventory.