Why isn't magnetic flux a dimensionless quantity?

I've read magnetic flux being described as "number of magnetic field lines passing through an area"

That is probably not a very good source to be learning from. This source basically has it backwards. You don’t start with field lines, count them, and get the flux. Instead you start with the flux and then draw field lines such that the number of field lines per area is roughly proportional to the flux. The proportionality is rough since you only draw a finite number of lines.