what is the minimum amount of unique shapes that could tessellate a plane?

What exactly are you asking?

Tessellation of the plane with 'shapes' requires an infinite number of them.

There are infinitely many ways to do that.

The shapes may or may not be polygons.

If they are polygons, they may or may not be regular polygons.

Polygons or not, they may or may not all be congruent.

If they are regular polygons, all congruent, they must be either equilateral triangles, squares, or regular hexagons. In the first two of these, the tessellation allows "sliding" of entire rows of them to make a new tessellation; with regular hexagons, this isn't possible, because hexagonal tessellations ("honeycombs") have no "fault lines."

If they are regular polygons, not all congruent, there are many ways to do that. If all of them that have the same number of sides are congruent, there are lots of those.

If the polygons don't have to be regular, still more possibilities open up. Some of the more interesting are the Penrose tilings.

If the shapes don't have to be polygons, there's another big increase in possibilities (check out some of M. C. Escher's work).

For pictures of a few of the astounding array of possibilities, see the Source: