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 is 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.

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.

The number of field lines is not a meaningful physical quantity, but only a useful tool to visualize the magnetic of electric fields. When I say that it is not a meaningful quantity is because it is not measurable, exactly because, 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.

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 is 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.