Does the spell darkvision count as magical darkvision? If the BBEG cast darkness, and one of the PC's casts darkvision, are they able to see through the darkness spell?
No, the PCs can't see through the darkness
The spell darkvision simply gives the PC darkvision just as if it were naturally granted by some other feature.
You touch a willing creature to grant it the ability to see in the dark. For the duration, that creature has darkvision out to a range of 60 feet.
And the darkness spell specifically says that a creature with darkvision cannot see through it. It doesn't matter if it is magically granted or not.
A creature with darkvision can't see through this darkness, and nonmagical light can't illuminate it.
So, no, a PC with darkvision (no matter how it is granted to them) cannot see through darkness from the darkness spell.
Note also that there's no such thing as magic darkvision or mundane darkvision in terms of the effects it has. The only difference is in how it is granted.