Atom, the omnipresent particle that builds the universe, hides its identity in a tiny, sub-atomic particle- the proton, where the proton number decides the type of the atom. However, an outer-nuclear component- the electrons- determines an atom's reactivity.
In nuclear reactions, the proton number can change so that the atom's identity also changes. However, in organic chemical reactions, only the electron count changes without affecting an atom's identity.
In a neutral atom, the number of protons in the nucleus is always balanced by the number of electrons in the outer nuclear region.
Ions are formed when an atom loses or gains electrons to attain greater stability and to form ionic bonds. If an atom gains an electron, there is an excess negative charge in the outer nuclear region that is greater than the number of protons in the nucleus. Such an atom is called an anion, and the excess charge is indicated with a negative sign in the superscript. For example, Cl-.

On losing an electron, the balance between the electron and proton again shifts so that there is an excess of positively charged protons. This excess is indicated using a positive sign in the superscript next to the number of electrons lost (or by how many the proton number is excess). Such a positively charged atom is called a cation. For example, Na+.

Now, ions can be monoatomic- made of one type of atom- like Na+ or Cl-. Or it can be polyatomic, where a group of atoms are held using covalent bonds, and one of the atoms loses (or gains) covalent bond electrons to become charged. For example- CO32-, NH4+.
Author Notes
Polyatomic ions are mostly formed by the heterolytic bond cleavage. There is still another type of bond breaking, which is impartial, known as homolytic cleavage. If heterolytic cleavage generates ionic species, homolytic generates radical species (read their comparison). Also, by calculating formal charges, there is a way to know which atom that is a part of the polyatomic ion is contributing to the maximum charge.
In organic chemistry, the most popular carbon-containing ions are carbocation and carbanion. These ions are less stable and always exist as transient species. These are generated in reactions as intermediates, so instead of a single-step reaction, the reaction proceeds in multiple steps.
Remember that
In organic chemistry, there are three main types of reactive intermediates - carbocation, carbanion, and free radicals.
So, after reading and understanding how ions are formed from an atom and also how an atom's identity can only change through nuclear reactions, we are quite capable of answering the question—are ions still an atom?
If you're still unsure, think of an orange. An orange without its peel is still an orange since an orange's identity lies within its seed.😊

Atoms, Ions and their Bonds are covered in detail in CurlyArrows' Introductory Organic Chemistry Course. Preview the Book.
Related Reading- Difference Between Cation and Anion
