Map with multiple scales

Yes, maps have different scales at different point, and, at a point, in different directions.

If you want a map with a constant scale along each parallel of latitude, then you want a Cylindroid map. That's a map that's either cylindrical or pseudocylindrical.

Cylindroid maps have a constant scale along each parallel. And each parallel is a straight line, all of which are parallel to eachother.

Cyindrical maps also have straight meridians, perpendicular to the parallels.

Pseudocylindrical maps are Cylindroid maps that aren't Cylindrical maps.

Cylindrical maps have many advantages. e.g. They treat all longitudes equally, and they're conformal (have perfect shapes) all along two parallels, whereas pseudocylindricals are only conformal at two points.

But Pseudocylindricals are very popular too, and often or usually have better shapes at high latitude, central-longitude positions than a cylindrical would. Many people would prefer the appearance of a pseudocylindrical, which might look more realistic. ...Well, they usually show Greenland better. But Behrmann Cylindrical Equal-Area is otherwise hard to beat by any other equal-area map.

Popular pseudocylindricals are Mollweide, whose grid is drawn in an ellipse (introduced in 1805), Eckert IV (introduced in 1906). Both of those are equal-area maps.

The Sinusoidal is another equal-area Pseudocylindrical. It's the simplest-constructed and easiest-explained:

Each parallel on some globe is drawn on a flat paper, with the same length that it has on the globe. The parallels are all parallel to each0ther on the paper (the Sinusoidal is pseudocylindrical). The result is a world-map that has a rather diamond-shaped look. It has relatively a lot of shape-distortion at its outer edges, but that's reduced if the Earth is mapped by two Sinusoidal maps, each showing half of the Earth. It's great advantages are its simplicity of construction, and its many kinds of scale accuracy:

The parallels are all of correct length (for a given globe), with the same scale along each parallel. The parallels are spaced as they are on that globe. The scale along the central meridian is the same as along the parallels.

That's a lot of correct things.

The Peters Projection (Gall Orthographic Projection, 1855) is a very popular cylindrical projection, and is also equal-area.

The Mercator for a long time was the most popular cylindrical, due to its excellent shapes. Mercator is conformal at every point on the map. A map is conformal at a point if, at that point, the scale is the same in every direction. Mercator, as a conformal map, is conformal at every point.

Mercator was the first map designed for conformality, introduced in 1569.

Mercator was also designed for a property that only it possesses. A line of constant compass-direction is a straight-line on Mercator, greatly facilitating the sailing of a reliable course between two points on the Earth.

An excellent equal-area Cylindrical is the Behrmann projection. It's equal-area like the Peters, but its shapes are better. Africa is shown far too skinny on most equal-area world-maps, but it looks fine on Behrmann.

In fact, Behrmann shows 2/3 of the Earth with as good shape as Africa, and scale at least equal to the scale along the equator--something that other ordinary Cylindrical Equal-Area maps can't do.

Behrmann is under-appreciated & under-used. Behrmann & Sinusoidal each beat the other equal-area maps in some worthwhile ways.
In fact, given that Behrmann, as a Cylindrical map, treats all longitudes equally, and is conformal all along two parallels (the 30th parallel, north & south), there are ways in which Behrmann beats all the other equal-area maps.