There are two factors:
- Presenting an alternate deity-like figure is an effective way to meet his goals
- He's starting to lose it a bit
The crucial part is earlier in the letter:
[Sauron] had gone the way of all tyrants: beginning well, at least on the level that while desiring to order all things according to his own wisdom he still at first considered the (economic) well-being of other inhabitants of the Earth. But he went further than human tyrants in pride and the lust for domination, being in origin an immortal (angelic) spirit. [...] Sauron desired to be a God-King, and was held to be this by his servants; if he had been victorious he would have demanded divine honour from all rational creatures and absolute temporal power over the whole world.
The Letters of J.R.R. Tolkien 183: Notes on W.H. Auden's review of The Return of the King. 1956
Sauron almost certainly didn't believe he was really Morgoth; in any case, Tolkien says in other writings that he was a professed atheist, but not a true (non-)believer:
Sauron was not a 'sincere' atheist, but he preached atheism, because it weakened resistance to himself (and he had ceased to fear God's action in Arda). [...] To wean one of the God-fearing from their allegiance it is best to propound another unseen object of allegiance and another hope of benefits; propound to him a Lord who will sanction what he desires and not forbid it. Sauron, apparently a defeated rival for world-power, now a mere hostage, can hardly propound himself; but as the former servant and disciple of Melkor, the worship of Melkor will raise him from hostage to high priest.
History of Middle-earth X Morgoth's Ring Part 5: "Myths Transformed" Chapter VII: "Notes on motives in the Silmarillion" (i)
Tolkien is writing this from the perspective of Sauron's time as a prisoner in Númenor, where he wanted to corrupt the Eru-fearing population; presenting an alternative deity is the best way to do that, and Melkor is the best choice - Sauron himself is hardly in a position of power.
After his "escape", though, it's a different story; Sauron suddenly is in a position of power (more so in the Third Age, when he was presumed destroyed and then returned - that's some serious God-cred), so why wouldn't he set himself up as a God-figure?
The specific choice of impersonating Morgoth is, so far as I know, never explored in any writings other than Letter 183, so a word-of-Eru answer is hard to come by; it does, however, present a certain logical sense. Morgoth is still a well-known and well-feared entity in Middle-earth, and most likely even more so in Harad and the East, where Morgoth dominated at least some of the population. It just makes sense.