Can a system of equations containing only 2 lines have exactly 2 solutions?
No - can't have exactly 2 solutions.
Can have 0 solutions (the two lines are parallel), 1 solution (the two lines intersect), infinite solutions (the two lines are collinear)
There are a number of ways to explain this, but I'll ask that you bear with me for a more tactile explanation. Take a minute and get two straws or pencils - two things that are straight, narrow, and easy to work with.
First hold them so that they touch - that is one point of intersection. There are many (and if we wanted to get into it, an infinite number) of ways we can arrange these two straws so that they touch in one place.
We can hold them so that they will never touch. One way to do that is to lay them on a flat surface near each other, and let's say both pointing left/right. Now the pencils are parallel and even if you were to have pencils that were super long (and we can say infinitely long), they will never intersect.
We can also place them so that every point on the one straw is touched by the other straw - this is called collinear .
Now, without bending them, can we make the two straws touch in only two places? The answer is no.