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I am studying electromagnetic theory and when I started researching the history of conventions used in magnetic interactions I could not get them. The basics of how they modelled the magnetic interaction are a bit confusing.

Like what is "number of magnetic field lines" trying to convey when field line are just a simple visualisation tool and approximation of the underlying magnetic vector field.

And why further concepts are build using this visualisation tool idea of field lines?

Like if field lines are just for visualisation why do measuring quantities such as magnetic flux and magnetic flux density and magnetic flux intensity and magnetic field strength depend on the amount of those field lines? I can draw how many ever field line I want right?

And how to correlate these measuring quantities such as magnetic flux and magnetic flux density and magnetic flux intensity and magnetic field strength with respect to the Vector field?

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"Magnetic force lines" is a misunderstanding. The truth of "magnetic force lines" is 0 magnetic moment lines, but not magnetic field force lines. A very simple method to judge it: take a magnet in each hands, make two magnets parallel, and feel the force between them, it is on the line that connect the center of two magnets, but not on the "magnetic force line", the angle between these two direction is 90 degrees.

If you still can't understand the "magnetic force lines",you can see the video [the truth of magnetic moment line/magnetic field force line][1] [1]: https://www.youtube.com/watch?v=oh19XoAR1n4&t=1s

And it is sure that there are some misunderstandings in current magnetism, these misunderstandings take strong confused, you need see they clearly, or to find the correct theory,I recommend this article: 场原理的理论分析——磁场 (第一部)The Theoretical Analysis of the Field Principle - Magnetic Field (1th piece) (https://phyyz.kinja.com/the-theoretical-analysis-of-the-field-1828705659?rev=1589649761587)

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As you point out, field lines exist only as a visualization tool. On a carefully drawn sketch, the direction of the lines indicate the direction of the field in any small region. At any point in the field, you can assign a vector. The vector would be in the direction of the field (or the lines), and the magnitude of the vector would indicate the strength of the field at that point. On a very carefully drawn three dimensional sketch, the density of field lines is proportional to strength of the field. This provides a way of visualizing flux. (Lines/Area)x(Area) = Lines.) Flux is field times area, which is proportional to the number of lines crossing the area in question. (The constant of proportionality is determined by the person drawing the sketch.)

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