soft question - Are there situations when regarding isomorphic objects as identical leads to mistakes?

First of all, as other posters have said, you never claim that two objects are identical unless they actually are equal as sets. However, there are times when it is appropriate to "identify" two isomorphic objects. (This is really just an issue of language).

Exactly when such identification is appropriate is a matter of some subtlety. If there are many different isomorphisms between two objects, and no natural way of selecting one of them, then it is generally a bad idea to identify them (although sometimes this is done implicitly, as when we talk about the field with $q$ elements). This applies in particular to a vector space and its dual. Generally speaking, if you have a unique isomorphism, or even a "natural isomorphism" (this can be made precise using category theory if desired), you are safe identifying them; however, this might conceivably produce problems if they are distinct subsets of some larger set you are interested in.

To further complicate the matter, in many, or even most, cases, we don't actually care about the identity of the set in question, only its (external or internal) structure. Thus, for instance, any questions about the real numbers that cannot be answered using the fact that they form a complete ordered field (e.g., is $1 in 2$?) are not really questions about the real numbers at all. This produces interesting issues in philosophy of mathematics; see this website for a discussion. Things get even more interesting when you start defining objects (e.g., tensor products, limits, colimits) using universal properties; in this case, the objects are only defined up to natural isomorphism, so there is no single construction that "is" the desired object.

There are also times when either of two identifications may be appropriate, but not both at once. For instance, $mathbb{Z}$ is naturally isomorphic to a fixed subset of the p-adic numbers $mathbb{Z}_p$, and to a fixed subset of the real numbers $mathbb{R}$, but no one would ever write $mathbb{Z} subset mathbb{Z}_p cap mathbb{R}$ (except to discuss when it is not appropriate to make identifications).

In short, it's a subtle question, and the answer is based more on experience than on any codified set of principles.