Thank you for taking the time to respond, Sergei. However, unless I misunderstand, this does not solve my problem. I need to apply this rule to every row in the sheet and have both cells affected by the conditional format. Thank you again! :)
I found a solution! (In case anyone else has this problem in the future and stumbles upon this post)
First, highlight A1:B1. Apply conditional formatting with the rule type "Format only unique or duplicate values". Select unique and format as desired.
Then, to apply en masse, but have them function independently, highlight A1:B1 again and double click the Format Painter. After that, hold the down arrow until all desired rows have had the rule applied to them.
If anyone knows of an easier way, please let me know!