The reason that this question even comes up is that there are "isotopes" of atomic species that have a slightly different atomic mass, but they have the same charge properties and bonding properties. These isotopes have different numbers of neutrons in their nuclei which carry no charge, yet have mass.
If by "identical" you mean that the different atoms of the same element have the same chemical bonding properties, then this is still true.
If by "identical" you mean that they have the same mass, then you may have a problem. However, we deal with this by sub-speciating the elements. There is a carbon-14 species, for example and a carbon-12. The small amount of carbon-14 that is generally found mixed in with carbon-12, makes it so that the molar mass of carbon is found to be 12.011 grams, and not just 12 grams.
Factor in this subspecies concept and the idea of "element" still works out just fine.
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