I am adapting part of this from an older answer but with more of a focus on the theological aspects of this question.

The "godlike status" of Egyptian Pharaohs should not be overstated, although on a religious and ideological level they were considered to be the divine incarnation of Horus they were not necessarily all-powerful or viewed that way by those around them. Arguably, the Pharaohs who reigned throughout Egypt's Dynastic period did not necessarily wield more power than the Ptolemaic kings, if anything the New Kingdom saw the increased development of administrative and infrastructural powers outside of the king, the most notable being that of the temples. The image of an all-powerful, divine king ruling Egypt absolutely is borne less out of the reality of Egyptian government and society and more out of a romanticised picture of a despotic, wealthy land that has been reinforced by the fact that the majority of surviving art and literature was designed to promote the ideal of kingship on a divine and symbolic level (i.e. Temple reliefs and statuary, and other commemorative contexts). While on an ideological and theological level the Pharaoh was both divine and human, he was also considered and described as a mortal and the very real limits of his power were based not in his authority but in his ability to exert power over the land, and especially the people. Regional and local elites and authorities for the most part carried out their duties with a level of autonomy while being directly incentivised and influenced by the king. The complicated Ptolemaic bureaucracy was built out of and around existing methods of bureaucratic control, these bureaucratic mechanisms by their very nature both enabled the Pharaoh to make use of the resources within Egypt while taking a measure of direct control out of his hands. Later Egyptian Pharaohs, like Psamtik, Nectamebo and even Ramses II ruled an Egypt that was in many ways more similar to that of the Ptolemies than that of the Old Kingdom Pharaohs.

Of course, equally important to assessing the realities of the position held by the Ptolemies, is understanding how they viewed themselves. The Ptolemids did assume the Greek title of Basileus or king and the corresponding feminine title Basilissa on coinage, statuary and Royal decrees in Greek and outside of Egypt they assumed the role of a Hellenistic king without exception. However the Greek influence actually has little to do with the Ptolemaic concept of queenship which was heavily influenced by Egyptian precedents.

Whether this makes it a straightforward continuation of Egyptian practice is similarly unclear, but I would generally consider it has having organically and independently developed by the Ptolemaic dynasty by drawing on examples from Dynastic Egypt. From the dynastic cult, which equated Ptolemaic rulers with Greek and Egyptian deities and deified them posthumously, and the co-rulership established by Ptolemy II and Arsinie II, and the precedent of Alexander the Great and Ptolemy I before them in being crowned Pharaoh, the Ptolemaic dynasty very heavily relied on Egyptian political and theological rhetoric and tradition to maintain their rule and as time went on the relationship between the Ptolemaic rulers and the Egyptian elites seems to have only grown stronger. The Ptolemids identified with Egyptian deities such as Amon-Re, Osiris, Horus or Isis and Hathor, as well as claiming divine descent. Although by this time these deities were equated with their Greek equivalents and sometimes Hellenised in depictions, they were also depicted frequently in traditional Egyptian form.

Cleopatra in particular identified with the Egyptian goddess Isis who was by this time considered the queen of heaven and was the divine consort of Osiris and mother of Horus. You can see her depicted as Isis in this full length statue of Cleopatra as the goddess Isis. In addition to this she was also associated with the Egyptian fertility goddess Hathor. The identification with Isis did extend to the Hellenistic Near East and Mediterranean where it was conflated with the deities Aphrodite, Venus, and Ishtar but again by this time the Egyptian pantheon was seen as analogous to Hellenistic and Roman pantheons in many respects. (See also Cleopatra depicted making an offering to Isis ). Cleopatra was considered a divine ruler in the sense that all Pharaohs were, and was technically the divine co-regent and consort of Ptolemy XIII, Ptolemy XIV, and Ptolemy XV. Her relationship with Ptolemy XV (Caesarion) took on a special significance because it was compared and equated with the relationship between Isis and her son Horus which is shown on coinage and on the famous temple scenes at the Dendera Temple Complex depicting Cleopatra with her son and heir Caesarion in traditional Egyptian style..

Essentially Cleopatra was divine in multiple ways, she was the Pharaoh who was the instrument of the gods on earth, she was a descendant of gods through her deified Ptolemaic ancestors, she was supposedly the living reincarnation of the goddess Isis who was the wife of Osiris and mother of Horus in the Egyptian pantheon, and in many ways she was treated as a deity in her own right under epithets like Neotera Thea ( the Youngest Goddess) and Thea Philopator. She was most certainly worshipped in connection to the dynastic cult like all the Ptolemies were, and her personal cult held a degree of popularity around the Mediterranean as a syncretised version of Isis/Aphrodite, Isis/Venus and various Near Eastern goddesses. Of course, not all Egyptians would consider her a literal god because personal beliefs varied, but as far as the temples and sanctuaries were concerned she was a bona fide divinity. In Egypt she also often assumes many of the roles of Hathor the heifer-goddess of fertility which makes sense given that Isis came to usurp many of Hathor's traditional role in later Egyptian traditions.

As time went on, the faith in the Pharaoh's divinity waxed and waned but with Cleopatra's reign, ironically one of the most successful uses of divine ideology in Ptolemaic history, it was about to fade away forever. Although Cleopatra's status as a deity did not disappear upon her death, indeed her suicide served as yet another form of apotheosis, Egypt was about to change forever with its integration into the Roman Empire. Despite the relative distance between the Ptolemies and their subjects, they were far mote culturally accommodating than the Roman Emperors who left nearly all Pharaonic duties to their subordinates and who rarely expressed a strong personal interest in Egypt's cults. The deification of the Pharaoh was absorbed into the Imperial cult but here it transformed further, the more traditional cosmic roles being relegated to gods like Osiris and placing posthumously divinified Emperors among the minor associated deities in other gods' temples while the Emperors' continued to be depicted as Pharaohs on temple motifs and were offered to and invoked in prayers. In this sense, the practice of viewing Egypt's ruler as a god did not end until the 4th Century and the adoption of Christianity and the dismantling of the Imperial cult and Egyptian paganism.