Why is it okay to say two groups are the same, when they're really isomorphic?

It is the central insight of mathematics of structures that things that seem to be different are "really" the same thing. It doesn't help understanding if you are not able to switch the usage of "same" according to context. After all, each pair of objects is different, so what would 1+1 actually mean? And no, it is not a good idea to try to do mathematics as a formal game without any meaning.

Because the real object is only defined up to an isomorphism. The thing you think is $G$ is really just a model, as is $G'$.

It is a question of language and of ontology (the essence of what a thing "is").

An Isomorphism can be seen as making a connection between two different aspects of the same abstract reality, thus illuminating its "true" nature more fully than either aspect.

Alternatively:

An Isomorphism can show us that two distinct entities with their separate realities look the same when seen from a particular perspective.

The concept of isomorphism does both jobs - hence the ambiguity - which is rarely a problem, unless you can give a specific example ... ?

Something I find useful is to think about abstract groups and realizations of each of them.

You could say that an abstract group is just a collection of objects with particular relations holding among them, defined through the group product (whatever it may be). You can completely specify an abstract group by writing down the multiplication table or its presentation (not an easy task, generally speaking).
Following this approach, you can say that

• there is only one group of three elements: its elements are $e,a,b$, where $e\cdot a = a\cdot e = a$, $e\cdot b = b\cdot e = b$, $a\cdot a = b$, $a\cdot b=b\cdot a = e$, $b\cdot b = a$ (sorry if I didn't write nicely the multiplication table). Then its presentation is $\langle a\;|\;a^3=e\rangle$ and you call it "cyclic group".
• there are two groups of four elements: one has the presentation $\langle a\;|\;a^4=e\rangle$ (cyclic group); the other is given by $\langle (a,b)\;|\;a^2=b^2=(ab)^2=e\rangle$ (dihedral group).

Now observe that you can interpret the group of order 3, usually denoted by $\mathbb{Z}_3$, as the set $\mathbb{Z}/3\mathbb{Z}$ of residue classes with addition$\mod 3$, or the set of the complex cubic roots of unity with the ordinary product over $\mathbb{C}$.
Similarly, the cyclic group $\mathbb{Z}_4$ is the group $\mathbb{Z}/4\mathbb{Z}$ of residue classes$\mod 4$, or the group of fourth roots of unity. The dihedral group $D_2$ is usually referred to as the group of simmetries of a segment (i.e. a degenerate regular polygon with two sides).

All of these groups, then, admit different realizations in terms of matrices: these are usually called group representations (e.g. you make use of rotation and reflection matrices to construct the 2 dimensional representation of every dihedral group). The sets of matrices used to build a representation are subgroups of $\text{GL}(n)$.

Every realization of a group is isomorphic to all the others. Hope this helps!

Isomorphism is just a renaming of elements.