If there are not, then would it be easier to say that 2 objects are identical as ordered fields as opposed to being isomorphic as ordered fields? Or is the word isomorphism used to emphasise the fact that the objects are different as sets?



Are there situations when regarding isomorphic objects as identical leads to mistakes?
Inside of the complex numbers there are lots of examples of distinct fields which are isomorphic. For instance, there are three subfields of the form ${\mathbf Q}(\alpha)$ where $\alpha^3 = 2$: take for $\alpha$ any of the three complex cube roots of 2 and you get a different subfield. What are the consequences of treating them as literally equal? You can't make any sense of Galois theory if you do that! Similarly, all $p$-Sylow subgroups of a finite group are isomorphic (since conjugate subgroups are isomorphic groups), but it would kind of destroy a lot of the content of the Sylow theorems by 
trying to say the $p$-Sylow subgroups are identical. 


More generally, anytime you have isomorphic but unequal objects inside a larger object, it can lead to confusion if not outright incomprehensibility if you try to regard them all as identical. (There was a paper by Chevalley about unit groups in number fields where he made a genuine error by an abuse of the "square root" notation and I think one might be able to express the mistake in the form of an isomorphism being confused with an equality, but I'd have to look at the paper again to be sure about this.) 


The word isomorphism does not emphasize that two objects are different; any group or vector space admits an isomorphism with itself using the identity map. The word emphasizes that in a structural way the two objects look like each other even though they are not literally the same. Never say two objects are identical if they are not actually identical. Having said that, I must admit that in mathematics one meets phrases like "since $X$ and $Y$ are isomorphic we can identify $X$ with $Y$" and then $X$ is replaced with $Y$. The usefulness of doing this depends on the application you have in mind. Note, however, that replacing $X$ with $Y$ is not saying that $X$ and $Y$ *are* the same thing. 


This question sounds like it is being asked by someone who hasn't had a lot of experience with isomorphisms and is trying to get a feel for what it means. In a year or two, after seeing more appearances of the concept and its uses, you'll get a better feel for it, but for now do *not* think the word isomorphism is a synonym for identical. 


