What you have hold of there is representation theory.
It's a lot easier to do this with something that has a physical interpretation simpler than a matrix. So I will do that first. Then I will do one example of a matrix.
Consider a 2-Dimensional system describing location. It is based on x and y coordinates. So you have the origin (0,0) and you have the set of points (x,y) representing the 2-D plane.
A location in this plane, say (3,4), is thus separated from the origin by a vector. We could call this vector P1 for position 1.
Now suppose we rotated our coordinate system by 90 degrees. The coordinate system, note. Not the points in the plane. The coordinate system is a mapping we put on top of this set of points. We now take every (x,y) point and in the new system, we have the new X axis pointing "up" along the old y axis. And the new Y axis pointing in the negative of the old X axis. So the old y value becomes the new x value. And negative of the old x value becomes the new Y value.
Apologies for the crudity of the diagram. It is intended to show that there are two coordinate systems overlaid on the same set of points. That is, we have the same point in two different (one rotated from the other) coordinate systems. It's the same point but its is represented in two different coordinate systems.
So, in the old coordinate system we would say P1 = (3,4). And in the new one, P1=(4,-3). That is, there is an actual point in the (x,y) plane, and there are different representations of it. We need to specify which representation we are in for the equality sign to hold.
Another way to say this is, the point P1 is an arrow that goes from the origin to the point. And it is represented by an ordered pair of coordinate values. The arrow does not change when you rotate the coordinates, but the coordinate values do.
So what about your matrix? There is a corresponding coordinate system, depending on what system your matrix fits into. Consider rotations. You can write a rotation around the z-axis (say it's the vertical axis where you are) like so. So if you stood up and rotated around the vertical axis by an angle theta, that is how you could write it.
But it is not the only way you could write it. You could lie flat on your back, then while laying on the ground you could rotate the same theta. Then you could stand up by "rotating" youself 90 degrees. You would come back to your feet rotated by the same angle. So this weird little dance of lying down first, rotating, then getting back up, let's you represent the rotation in the x-axis. There it looks like so. It's the same rotation, just represented in a different coordinate system.
Generally this is what is happening with your matrix. In a particular representation there are particular numerical values. There is a mathematical object, the matrix, and in a particular representation it is equal to those numerical values in that arrangement.
Mathematicians have a way of presenting this where such things are mappings. So the point P1 is a mapping from the origin to a point in the plane. It is the arrow. Physicists usually want the coordinates, and so will talk about P1 as having x and y components. These two views are equivalent. That equivalence is why the equality sign is acceptable.