Shapes and numbers do exist and I believe they exist objectively.
Aren't you a realist, then?
But they are abstractions. Unlike constructs they don't go away if nobody believes in them, they can be rediscovered any time by a sufficiently complex mind and they will functionally be the same.
So, if a meteor strikes the Earth and wipes out all life, this changes the fundamental nature of right triangles?
The Pythagorean theorem is a truth-statement about the nature of right triangles. It says that the sum of the squares of the interior angles equals the square of the hypotenuse. THAT doesn't change if all beings capable of understanding such a principle cease to exist. The moon obviously exists. How do we know? We can see it. We can point to it and say, "Hey, there's the moon." But the Pythagorean theorem also exists. A mathematician can point to the relation between three points in space and say, "Hey, there's the Pythagorean theorem."
If all life were wiped from the face of the earth, there would be no one around to say, "Look, there's the moon." But the moon would still be there. Likewise, if all life were wiped from the face of the earth, there'd be no one to say, "Look, there's the Pythagorean theorem." But (just like the moon) it'd still be there. If a slime mold somehow survived the apocalypse and evolved into a thinking creature, and these creatures started thinking about right triangles, there it would be... the Pythagorean theorem.
I say the Pythagorean theorem is just as real as any material object. The only thing separating it from realness is an arbitrary ontology that classifies material objects as the only things that are real.