Under most conditions your companion will get sequential boarding positions with you. As you said, Business Select tickets are an exception. One person buying EarlyBird may help you both get better boarding positions and Anytime tickets include EarlyBird in the fare price. Anytime fare EarlyBird is no better than a Wanna Get Away fare with EarlyBird added so don't buy Anytime just to get EarlyBird. My usual recommendation is to check in like you normally would 24 hours prior to departure then if you don't like the boarding position purchase the upgraded boarding option through the app. You will need to purchase for both passengers though as the sequential boarding does not factor in when purchasing the upgraded boarding.
--TheMiddleSeat
Edit: Southwest does prioritize customers with EarlyBird based on their fare type. I incorrectly said Anytime EarlyBird is the same as Wanna Get Away EarlyBird. Here's the Southwest FAQ:
Customers who have purchased Anytime fares receive priority over Customers who purchase EarlyBird Check-In with Wanna Get Away Plus and Wanna Get Away fares. Customers who purchase EarlyBird Check-In with Wanna Get Away Plus receive priority over Customers who purchase EarlyBird Check-In with Wanna Get Away fares. Boarding positions are assigned based on the time stamp of the EarlyBird Check-In purchase relative to Passengers within the same fare product.
