Lets take a category of sets and functions between them. So there are (infinitely?) many terminal objects. As far as I understand, it's no so hard to prove that any two terminal objects are isomorphic to each other.
Concept of being "unique up to isomorphism" still seems a bit unclear for me, so I'm wondering wether it is technically correct to say that those singletons-terminal objects are "unique up to isomorphism". It feels like I can go further and say that those are unique up to unique isomorphism, because the definition a of terminal object implies that no other isomorphism can take place simply becase no more arrows in-between any two terminal objects can appear.
Your intuition is good. "Unique up to isomorphism," is usually weaker than what you want to show in most contexts in that there will also be a uniqueness statement for the isomorphisms themselves. In the case of terminal objects, you do literally get that there is a unique isomorphism between any two terminal objects, but in general you might not get something quite so strong.
Consider, for example, the cartesian product of sets $A \times B$. There are multiple sets satisfying this property, and they're all isomorphic, but there may be more than one isomorphism between them. In particular, any nontrivial automorphism of $A$ gives rise to an automorphism of $A \times B$ that is not the identity. The right uniqueness property for the isomorphisms, here, is that there is a unique isomorphism between any two realizations of $A \times B$ that commutes with the projections to $A$ and $B$.