Characteristics of pressure in liquid

The other answer is good. It is a function of depth only. In an atmosphere it could be also altitude (which is a depth). Just think of the column of fluid above it. Even if shape of the vessel is varying, just depth. The sideways forces balance out. Every tiny cube is holding itself with force going equally in every direction, plus a tiny addition for its weight. That’s why it increases linearly with depth (if constant density).

The hydrostatic pressure is the weight of the volume of liquid that the blob of liquid is holding up. That is - the weight of the liquid directly above the blob. The liquid a bit to the side is held up by another blob, and so does not count as pressure.

Note: the intuition that the liquid is pressing in on the sides is, in a sense, correct, but this is just the balance of forces. If this was not there, then the blob would squish out sideways. The value of the pressure (force per unit area) on the top surface of the blob has to be exactly enough to balance the weight above.

This can also be worked out by integrating the pressure over a hypothetical membrane that goes horizontally across the whole body of fluid to the container and then noting that the force of the bottom layer on the top layer must equal the weight of the top layer.

And keep in mind that if the container is, say, a cone, then the container itself is pushing up on the liquid as well. So, in the cone, the lower layer is not holding up the whole upper layer itself - you have to project this onto a hypothetical surface that is as large as the top of the cone.