When we say, that, say, a surface contains ${\infty}^{k}$ lines, do we mean that it contains a kparameter family of lines? Do we assume that this family is parametrized by a $P^{k}$, say, or we use this term more informally? This is certainly a standard notation, but I didn't see its explanation in standard modern textbooks.

The terminology is fairly common in classical works on projective algebraic/differential geometry. I am not aware of its origins. Anyway it is used rather informally and only means that you have a $k$dimensional "continuous" family of objects. The parameter space usually is only local and should be thought as a small ball in $\mathbb C^k$. 
