Can I have a game purchased elsewhere added to my Steam library?

It's not possible to add a non-steam game like you bought it from Steam. There is a page on linking non-steam games to Steam, but this will only generate a shortcut for the game in Steam. You won't be able to download updates:

This will only add a shortcut to your game in the Steam client. This will not allow you to download updates for your game through Steam, nor allow you to use this game as proof of ownership for your Steam account.

When you buy a game through Steam, you buy a licence to use that game on Steam alone. This is also the case with other similar gaming sites.

https://support.steampowered.com/kb_article.php?ref=2219-YDJV-5557

I want to add it to my library, but not as a non-steam game

I want it to be like I bought it from steam

This is only possible if you buy it on Steam or if it is available to you via Family Sharing from someone who bought it.

It isn't possible but you can Use GOG Galaxy to sync your games without minding the platform you bought it from.

Here is where you can download GOG Galaxy

It scans your PC for games and Launchers and then puts them and you can even add a game if GOG Galaxy wasn't able to detect it.. It's pretty cool and I even tried it.