You do not buy the software through Steam (or via SquareEnix), you buy game time access, as in the account..... So you do not need to buy it again, because you already have an account.
The game client is free.
Transfer the game install location files from the old computer to the new computer.
Copy the game saved data/screenshots from the older computer ( C:\Users\<Your computer user name>\Documents\My Games\FINAL FANTASY XIV - A Realm Reborn ) to the same location on the new computer.
Install the Steam Client on the new computer, then see:
https://gaming.stackexchange.com/que...eady-installed