Well I have a farm that I only dance off against which uses intel. Does strength even matter when I hit him because it's an intel action?
When you hit him/her, read the summary. The summary tells you how much intel and strength you used each hit.
When you're dancing some one, it relays on whose total intel is stronger. This takes into count your intel stat, your intel tb, your intel misc and of course pots. Who ever has the highest intel has the better chance of winning. Same works with strength. Str doesn't matter when you're dancing, intel doesn't matter when you're fighting.