"The property $P$ holds up to isomorphism" is a phrase which means "we might say an object $X$ has property $P$, but that's an abuse of notation. When we say that, we really mean that there is an object isomorphic to $X$ which has property $P$". Essentially, it means "the property might not hold as stated, but if we replace the idea of equality by the idea of isomorphism, then the property holds".

Relatedly, "The object $X$ is well-defined up to isomorphism" means "if we replace $X$ by an object isomorphic to $X$, we still obtain something which satisfies the definition of $X$."

Examples

Groups of order $2$

There is only one group of order $2$ up to isomorphism. We can define the object "group of order $2$" as "the group with two elements"; this object is well-defined up to isomorphism, in that while there are several different groups of order $2$ ^[1], any two such groups are isomorphic. If we don't think of isomorphic objects as being "different", then there is only one distinct group of order $2$.

1. ^{^︎}

   Two such groups are $0,1$ with the operation "addition modulo $2$", and $e,x$ with identity element $e$ and the operation $x^2=e$.