Yes, according to Advaita, there is no difference in Atman and Brahman.
Due to ignorance Atman tastes the fruit of Karma and believes him as occupying the subtle body but when he realize that I'm Atman, upon clearing the ignorance, became free from grief (bondage of samsara) and became grateful. So, Here one bird refers to Jiva with ignorance and other refers to realized Atman.Note1
Here is Adi Shankaracharya's commentary on following verses of Mundaka Upanihad:
द्वा सुपर्णा सयुजा सखाया समानं वृक्षं परिषस्वजाते ।
तयोरन्यः पिप्पलं स्वाद्वत्त्यनश्नन्नन्यो अभिचाकशीति ॥ १ ॥Note2
Com.—The Para vidya has been explained, by which the immortal ‘purusa’ or the Truth could be known, by whose knowledge the cause of Samsara, such as the knot of the heart, etc., can be totally destroyed. Yoga which is the means to the realization of the Brahman has also been explained by an illustration “taking the bow and the rest.” Now the subsequent portion is intended to inculcate the auxiliary helps to that yoga, as truth, etc. Chiefly, the truth is here determined by another mode, as it is extremely difficult to realize it. Here, though already [Page 65] done, a mantra (brief) as an aphorism is introduced for the purpose of ascertaining the absolute entity. Suparnau, two of good motion or two birds; (the “word Suparna” being used to denote birds generally); Sayujau inseparable, constant, companions; Sakhayau, bearing the same name or having the same cause of manifestation. Being thus, they are perched on the same tree (‘same,’ because the place where they could be perceived is identical). ‘Tree’ here means ‘body;’ because of the similitude in their liability to be cut or destroyed. Parishasvajate, embraced; just as birds go to the same tree for tasting the fruits. This tree as is well known has its root high up (i.e., in Brahman) and its branches (prana, etc..) downwards; it is transitory and has its source in Avyakta (maya). It is named Kshetra and in it bang the fruits of the karma of all living things. It is here that the Atman, conditioned in the subtle body to which ignorance, desire, karma and their unmanifested tendencies cling, and Isvara are perched like birds. Of these two so perched, one, i.e., kshetrajna occupying the subtle body eats, i.e., tastes from ignorance the fruits of karma marked as happiness and misery, palatable in many and diversified modes; the other, i.e., the lord, eternal, [Page 66] pure, intelligent and free in his nature, omniscient and conditioned by maya does not eat; for, lie is the director of both the eater and the thing eaten, by the fact of Ids mere existence as the eternal witness (of all); not tasting, he merely looks on; for, his mere witnessing is direction, as in the case of a king.
Advaita explanation get clear from Adi Shankaracharya's Bhashya on next verse:
समाने वृक्षे पुरुषो निमग्नोऽनिशया शोचति मुह्यमानः ।
जुष्टं यदा पश्यत्यन्यमीशमस्य महिमानमिति वीतशोकः ॥ २ ॥
Com.—In this state of things, the Jiva, i.e., the enjoyer occupying the body as above described, under the heavy load of ignorance, desire and thirst for the fruits of Karma, etc., sinks down like a bottle-gourd in the waters of the sea, is convinced beyond doubt that the body is the atman and thinking that he is the son of this man or the great-grandson of that, lean or stout, with or without good qualities, is enjoying or suffering, and that there is none but him, is born, dies, is united with and parted from relations and kinsmen; therefore, he grieves from helplessness thus: “I am good for nothing; I have lost my son; my wife is dead; what [Page 67] avails my life” and so forth and is subject to anxiety from ignorance owing to numerous kinds of troubles; but when thus constantly degenerating in births, of pretas, beasts, men and the like, he happens, owing to the result of pure deeds stored up in many (previous) births to be instructed in the path of Yoga by some preceptor surpassingly compassionate and being qualified by abstinence from giving pain, truth speaking, continence, complete renunciation and control over the internal and external senses and with his mind concentrated, finds by dint of meditation, the other who is approached by different paths of Yoga and by the followers of Karma distinct from him, conditioned in the body, not subject to the bondage of Samsara, unaffected by hunger, thirst, grief, ignorance, decay and death and lord over all the universe and thinks thus: “I am the atman, alike in all, seated in every living thing and not the other, the illusory atman, enclosed under conditions created by ignorance and this glory—this universe is mine, the lord of all,” then he becomes absolved from grief, i.e., is released entirely from the ocean of grief, i.e., his object is accomplished.
Note1: Actually both the Jiva and Ishwara (both birds) are illusive title due to ignorance, and in absolute there is only Brahman.
Note2: This verse is found in both Mundaka and Shvetashvatar Upanishads and looks actually comes from Rigveda 1.164.20.