False ego means accepting this body as oneself. When one understands that he is not his body and is spirit soul, he comes to his real ego. Ego is there. False ego is condemned, but not real ego. In the Vedic literature (Bṛhad-āraṇyaka Upaniṣad 1.4.10) it is said, ahaṁ brahmāsmi: I am Brahman, I am spirit. This "I am," the sense of self, also exists in the liberated stage of self-realization. This sense of "I am" is ego, but when the sense of "I am" is applied to this false body it is false ego. When the sense of self is applied to reality, that is real ego. There are some philosophers who say we should give up our ego, but we cannot give up our ego, because ego means identity. We ought, of course, to give up the false identification with the body.
(Bhagavad-Gita---13:8:12--purport).

The conditioned soul thinks, "I am this body," but a liberated soul thinks, "I am not this body. I am spirit soul." This "I am" is called ego, or identification of the self. "I am this body" or "Everything in relationship to the body is mine" is called false ego, but when one is self-realized and thinks that he is an eternal servitor of the Supreme Lord, that identification is real ego. One conception is in the darkness of the threefold qualities of material nature—goodness, passion and ignorance—and the other is in the pure state of goodness, called śuddha-sattva or vāsudeva. When we say that we give up our ego, this means that we give up our false ego, but real ego is always present. When one is reflected through the material contamination of the body and mind in false identification, he is in the conditional state, but when he is reflected in the pure stage he is called liberated. The identification of oneself with one's material possessions in the conditional stage must be purified, and one must identify himself in relationship with the Supreme Lord. In the conditioned state one accepts everything as an object of sense gratification, and in the liberated state one accepts everything for the service of the Supreme Lord. Kṛṣṇa consciousness, devotional service, is the actual liberated stage of a living entity.
(Srimad Bhagavatam---3:27:13---purport).

Although the living entity's mind, intelligence and identity are beyond the range of this material world, when he enters into this material world due to his desire to dominate matter, his original mind, intelligence and body become covered by the material energy. When he is again uncovered from these material or inferior energies, he is called liberated. When he is liberated, he has no false ego, but his real ego again comes into existence. Foolish mental speculators think that after liberation one's identity is lost, but that is not so. Because the living entity is eternally part and parcel of God, when he is liberated, he revives his original, eternal, part-and-parcel identity.
(Teachings of Lord Caitanya).