Axes are relatively simple tools to make: a blunt, preferably sharp/narrow rock attached sturdily to a handle. How were knives made in primitive or native civilizations without metalworks or blacksmiths?

Knives would be made the same way as axes, by "flintknapping" or reducing the excess stone on a large piece of quality material (like chert or flint) until you have a sharp cutting edge. The only difference between a stone axe blade and a stone knife would be the shape of the object and the thickness of the edge.

Most often, a single "flake" taken from a "core" of chert or other high quality stone will be used as a "blade". These blades would serve the function of knives, having a sharp cutting edge. You could also create a more formal knife, but usually that process was reserved for creating spear/arrow points and axes. Obsidian in particular makes for very sharp blades.

As for axes, a handle is not always a requirement. Hand axes are a very popular form, often in a teardrop shape where the wide end is the handle.

I can personally attest to the cutting power of these sorts of knives. I've certainly used metal kitchen knives that didn't cut as well. If quality stone is used to make the knives the edge can be incredibly fine. The big advantage of metal tools over stone is in durability. The edge wears down on the stone tools much quicker.

Edit: A wooden or bone handle can be attached to the stone blades to create a more typical knife, but again, like the hand axes, a handle is't strictly a requirement.