I am sure this question was asked here like may be hundreds time, but still let me ask once again:
Why da hell SWTOR's Sith Empire have Republic LOGO from 1-3 episodes
For those who might not be aware, Palpatin simply didn't changed the LOGO, then he transformed Republic to Empire. That was understandable political choice, since he tried to show like he actualy saving the Republic.
But i hardly can imagine that Republic itself had copied old Sith Empire LOGO (after events of SWTOR era)
Not entirely sure; however, a safe guess would be that Palpatine used this symbol to unite the Republic as it is connected to Ancient Sith tradition and Palpatine is secretly Sith. I would gather that it wasn't a dead giveaway for Palpatine's motives because no one probably remembers it as a Sith symbol, being that it was last used thousands of years ago. I actually do not remember seeing this symbol until at least episode II where the clone army's existence is uncovered and Palpatine's plot is set into motion. However, I would guess that the Republic has the Sith Empire logo and not vice versa by these estimations. I don't think the Republic did copy a Sith Empire logo, it was probably Darth Sidious who was posing as Palpatine that set that into motion as a sort of irony hearkening to the level of deception he employed.
If you're interested, there are numerous things in Star Wars that are based off of real cultures, places, and history. I.e. Budapest is a Star Wars city, it is a place in Russia also. Palpatine is akin to the word in Latin meaning "tongue." The Empire's symbol resembles the Japanese Empire's flag. We could go all day... powerful symbols from anciet cultures sometimes come back in real life too. And their origins as so old hardly anyone knows them.