Ultimately, it is just a matter of convention and how a particular author wants to write things. Authors usually choose the path that makes their writing more succinct and easier to read.
The example you gave is a common one: Make a product of objects... shouldn't the original objects be subobjects of the new object? Depending on your decision, the definition will be one or the other. For another example different from just ideals of rings, consider any big von Neumann ring $R$. The thing is filled with idempotents, and for each nontrivial idempotent $e\in R$, $eRe$ is again a ring (with identity $e\neq 1$!)
But let's also keep in mind that there are a great many arguments that do depend on sharing the identity, and choosing that convention is the best for them.
I think people often get too stressed about this thing with identities in particular. Just remember that definitions are (most often) not a matter of correct or incorrect but more of a matter of clarity and ease of exposition. One definition or notation is "better" than another if it is relatively easily understood and minimizes confusion. The mistake many people make is thinking that there is always a universally "best" definition/notation.