You are confusing a lot of things. Let me try to explain a bit.
PPI and DPI are not the same units. A Pixel is an information unit. The values of color on one location of an image. And a Dot is... a dot. A tiny droplet of ink.
Dots are not squares, rectangles, circles, or ellipses. They are droplets of random shapes dictated mainly by the absorption of the fibers of the paper.
One reason for the printers having different DPI on the horizontal axis, vs the vertical axis is that they are controlled by two different mechanisms. The X-axis is controlled by a moving printer head on a lane, or a laser beam shoot by a rotating mirror. The Y axis is controlled by rollers or a rotating drum. If you wanted the same DPI on both axes you would get the smaller number, because one mechanism is rougher than the other (600x600 print instead of 1200x600 for example). But you want smaller dots by overall area so if you can make them smaller even in only one direction, do it. They are so small that you do not see the shape.
for the entire page
No, they are Per Inch. That is what the second P and the I mean. 1200 dots each inch.
Why don't printers print square pixels by default?
A printer will print a pixel on whatever aspect ratio the image has. It has nothing to do with how many DPI they print.
Imagine these images are 5x5 pixels. You can squish or squash the image, so the pixels are no longer square. The printer will print them accordingly.
Here is a simulation of different DPI Two sets of checkerboard of the same size printed in different dot sizes. Simply means, smaller dots or bigger dots. (In this simulation I have little squares, but they will be random shapes on a real print)
And here is a simulation of two different PPI using the same image.
The first image has 5 pixels on one inch. The second one has 2.5 pixels on one inch.
Here is another simulation
In order to clearly see it, you need to open it on a new tab at 100% so it is not scaled inside this post on your browser. The image is optimized to be viewed on a Full HD monitor.
The first image has black and white dots of 1 pixel, simulating a print of "X DPI" Let me put an example number. 100 DPI.
The second image has twice bigger dots. It has X/2 DPI or 50 DPI. Each square is 2x2 pixels.
The third image has vertical rectangles as dots. It is simulating a resolution of, let's say 100x50 DPI. Each dot is 1x2 pixels.
If you move away from the screen, you will not really notice the rectangular dots, but you will appreciate the improvement in detail compared to image 2.