Why are the number of magnetic field lines finite in a particular area?

I find it very confusing when people say that the strength of the magnetic field is proportional to no. of field lines/area. Why is this terminology still used?

The terminology is still used because it is correct and it gives a graphical way to understand magnetic fields that is particularly intuitive to understand and apply.

For an example, consider the set of all lines in the plane passing through the origin and through the vertical line segment $x=1$ and $y=[0,1]$. There are an uncountably infinite number of points in that segment and a unique line for every such point. There are no gaps or missing points.

Now consider the set of all lines passing through the origin and the vertical line segment $x=2$ and $y=[0,1]$. Note that there is also an uncountably infinite number of points in this second segment and a unique line for every such point in the second segment. There are again no gaps or missing points.

Now, consider the relationship between these two sets of lines. All of the lines that go through the second segment also go through the first, but the reverse is not true. Half of the lines that go through the first segment do not go through the second. Therefore, indeed the number of lines through the second is less than the number of lines through the first. Half of an infinite number of lines is still an infinite number of lines, so the cardinality of the set is unchanged (infinity is weird).

No gaps have opened up, but the number of lines through the second segment is, in a physically valid sense, half the number of lines through the first segment. Sometimes the ratio of two infinite quantities is finite. Of course, we cannot draw every such line, but we can draw a representative set of a few and convey the concept of the whole field of lines. Doing so allows us to correctly and intuitively reason about the behavior of the field in a way that is difficult using the integrals directly.

This is the sense in which the strength of the magnetic field is proportional to the number of field lines/area. There are an infinite number of lines through each area, but some of the lines going through one area do not go through another. The proportion of lines that miss the other area is the proportion that the field strength decreases.

why there are gaps between the the iron filling lines?

Iron filings are ferromagnetic. They don't just show the field, they change it.

...hence the iron filings align themselves to stronger field lines.

The filings self-organize into distinct lines because their presence concentrates the field. Magnetic field lines prefer to go through a ferromagnetic body rather than through empty space. The field actually is stronger inside the iron particles than in the gaps between them.

If you drop a new filing into the gap between two of the visible "lines," it will feel attraction toward either of the surrounding lines. It will only stay put, and become the seed for a new line, if the magnetic force that it feels is too weak to overcome the static friction between the particle and the paper (or whatever) underneath.

This is a very under-discussed topic in introductory physics classes. The instructor would just tell you this for a fact with no further discussion on the topic which is the source of a lot of confusion. Well, as answered already, it is a matter of convention. You "agree" to draw a certain number of field lines and then compare, "fixing" the number of field lines to what you agreed. This concept is better understood when talking about electric fields. Know that electric field due to a charge $q$ is $$E= frac{1}{4piepsilon_0}frac{(q)}{r^2}$$

Field density is something you'd appreciate being called what we're dealing with. Let field density be defined as

$$D= frac{n}{A}$$

$n$ is number of field lines passing through any surface we choose and $A$ is area of that surface. For simplicity, sphere is the most symmetrical surface, so we choose it to be a sphere. Note that it is purely conventional. $r$ be the radius of sphere. We choose a sphere around a charge, certain number of field lines cross it, we choose a larger sphere, same number of field lines cross it, but now less "densely" namely, less number of field lines per unit area.

The convention is that we choose $frac{1}{epsilon_0}$ lines for a unit charge. A charge $q$ would "give out" $frac{q}{epsilon_0}$. This convention makes life simple as in, the Electric field at any point is now the lines density itself.

Similarly, for magnetic field, since we choose to draw a certain number of field lines, say 7 field lines for every one "bar magnet", then we can cleverly compare things without getting mathematically complex.

The number of magnetic field lines you draw are just a means to describe the strength of a magnetic field, so it is a convention to draw more lines.

If the field is stronger, but even if you draw only a few lines, the magnetic field between the lines still is there.

These lines do not really exist. They just show in what direction the force is. You could draw as many lines as you wish, but then you would not see anything any more.

The number of field lines is not a meaningful physical quantity, but only a useful tool to visualize the magnetic of electric fields. It is not a meaningful quantity because it is not measurable, for the reason that, as you said,

"One can draw/imagine as many unique (curved/straight)lines as he/she wants in some specified finite area (assuming that each line is unique if it doesn't overlap with another line)."

In other words, the number of lines is $N=a B$ where $B$ is the field and $a$ is a proportionality constant. However, the constant a is arbitrary, and you can basically decide how many lines to draw in order to make your plot/figure looking better. The number of lines is just a useful way to visualize the field, they are not a physically well defined quantity. Another reason why they are not physically well defined is because the number of lines is a discrete objects, but fields are continuous. Consider a uniform field with field lines parallel to each other. The field is constant at any point in space, but there are the white regions between field lines where there are, by definitions, no lines. These points also have a finite field, but zero number of lines. So, places where the no. of lines is zero have no special meaning, they do not have a field weaker that other places.

Also, consider that, practically, there is no place in the universe where the magnetic field is zero. In order to have no magnetic field you need 1) that the charge distribution is completely static in your reference frame (no currents), or that you are infinitely far away from any moving charge, and away from any source of propagating electromagnetic waves.

The terminology is only used to visualize the fields. Usually, advanced text books do not even mention the concept of number of field lines.