Why is positive (semi-)definite only defined for symmetric matrices?

When we are defining positive (semi-)definite matrices, we do so for symmetric matrices only.

Why do we need symmetry in the definition?

It's actually defined more generally for Hermitian matrices (matrices equal to the complex conjugate of their transpose). If a Hermitian matrix has at least one entry with a (nonzero) imaginary part, then the matrix is not symmetric, but it could still be positive (semi-)definite.

There are two reasons I can think of why we wouldn't extend the definitions to non-Hermitian matrices. The first is that if the matrix is not Hermitian, then we could have complex eigenvalues. In that context, "positive" has no meaning.

Another reason why I think we wouldn't has to do with quadratic forms. If $A$ is a real symmetric matrix, and $x$ is a vector of variables, then $f(x)=x^TAx$ is called a quadratic form. $f(x)>0$ ($f(x)\geq 0$) for all nonzero $x$ if and only if $A$ is positive definite (semi-definite). We could relax the condition that $A$ is symmetric, and define quadratic forms in the same way for a general real matrix. But the resulting quadratic form is the same as the one defined by the real symmetric matrix $(A+A^T)/2$. The quadratic form is positive definite if and only if $(A+A^T)/2$ is, so positive definiteness is not a property of $A$ as much as it is a property $(A+A^T)/2.$ In other words, we would end up studying symmetric matrices anyway, so wouldn't get anything new by extending the definition to non-symmetric real matrices.

Unfortunately, not everybody does it this way, which can produce some confusion. And in the case of complex matrices, what you want is Hermitian, not symmetric.

Basically the answer is that with the symmetry assumption, positive (semi)-definite matrices have some very nice properties, and without it they don't.

A simple intuition is that positive definite matrices are matrices which eigenvalues are all strictly greater than zero. If the matrix is not symmetric, it might not even have eigenvalues in $\mathbb{R}$.

(semi)-definiteness is usually associated with symmetric bilinear forms (and sometimes sesquilinear forms), and the Gram matrices for those forms are symmetric (or Hermitian, as they case may be.)

The condition can be studied outside of bilinear forms, but in that case it is particularly useful and natural.