Yes, the atomic number of an atom is equal to the number of electrons in a neutral atom. In an electrically neutral atom, the number of protons in the nucleus is equal to the number of electrons surrounding the nucleus.
For example, carbon has an atomic number of 6, which means it has 6 protons in its nucleus. In a neutral carbon atom, it also has 6 electrons orbiting the nucleus.
The atomic number uniquely identifies an element because it determines the number of protons, and hence the chemical properties of the atom. The arrangement of electrons in an atom's electron shells determines its chemical behavior, bonding characteristics, and overall reactivity.