There are many such solids:
A heptagonal pyramid
A cuboid (or rectangular prism)
A hexagonal dipyramid
A pentagonal pyramid with a 2-vertex shape on the other side
etc.
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A cuboid.
A rectangular prism has 8 vertices. A rectangular pyramid has 5 vertices.
An octohedron
The solid figure that has the same number of faces and vertices and has 8 edges is a cube. A cube has 6 faces, 8 vertices, and 12 edges, so it fits the description given.
A cube.