I need to do this for displacement mapping, subdivision gives a bad result and makes triangles distinguishable.
Related to: How to divide a circle into a square grid?
But I get the error "No other selected object found to use for projection".
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Knife project could help you:
subdivide a plane that will become your "shape cut" grid.
then over it (higer in Z) put you shape (just the outer edges, delete everything else)
then select - first the cutting shape - then (shift-select) also the background subdivided plane, then search and use "knife project"
then, your plane now has a cut shaped exactly as you want
just invert the selection,
and delete everyhing else
now of course you can extrude this mesh
If you leave the face in the shape and instead delete faces only from the grid (leaving the edges) you can reverse the selection of shape and grid and cut (knife project) the grid to the shape. Then the shape is ready and you don't need to cut off parts of the grid to get your desired result.