What is the difference between being unique, unique up to isomorphism and unique up to unique isomorphism?

What is the difference between being unique, unique up to isomorphism and
unique up to unique isomorphism?

Could anyone explain the difference between the above mentioned terms
"unique", "unique up to isomorphism" and "unique up to unique
isomorphism", preferably with an example? Are there other
characterisations of being unique "up to something"?