A lot of your question seems to be premised on a misconception. The Persians wouldn't have cared about being called "barbarians" because the Greeks weren't calling them "barbarians" in English where the word means "savage" or "uncivilized." Greeks called the Persians "barbaroi" (βάρβαροι) because it was simply the Greek word for "foreigner."

The Romans adopted the Greek word into their own Latin language, and through the influence of those two languages and their frequent contact with less settled foreign tribes, it eventually took on a negative connotation. When the same word was borrowed into English as "barbarian," it had a different meaning than originally intended by the Ancient Greeks.

The Greeks did engage in a level of proto-orientalism, both by fetishizing and demeaning aspects of Persian culture, but the word "barbaroi" wasn't part of that. Unfortunately, we don't have many Persian sources in general (something I've discussed before), so we do not know how they responded to the Greeks' views of them. It's worth remembering that Greece has an outsized influence on modern western culture. During the period of their wars with the Persian Empire, Greece was a fractured region on the fringes of settled civilization. The Persians had very little reason to worry about Greek opinions any more than the other couple dozen ethnic groups in their empire.

called Persian barbarians as they were ruled by a single monarch (in contrast to Greek republican states).

As you might have already guessed, the use of "barbarian" had nothing to do with what system of government anybody was using. However, it's still worth acknowledging that even if it had been intended as an insult, there was no widespread Greek animosity toward monarchies. There were still Greek cities in Thessaly, Libya, and Sicily with monarchs of their own. All over the Greek world, autocracies called "Tyrannies" sprung up. Once again, that's a word with a negative connotation today, but in ancient Greece to title Tyrannos just meant an autocrat who was not a king, and was treated as completely legitimate form of government.

And did either side accuse the other of slavery (even though we know they all had slaves)?

This is something that happened. As you said, both the Greeks and the Persians had slaves. The Persian Empire was huge and diverse. Different parts of the empire used enslaved people in different ways. In the west, closer to Greece, slave labor was more common on a larger scale. Further east, near the capitol cities like Babylon and Persepolis, "free" peasant labor remained more common.

Despite this, Greek authors like Herodotus regularly referred to the Persians defeating and enslaving the whole population of Greek cities and described how every subject of the Persian King was made into a slave. These issues are more complicated.

The Persians really did capture all or most of cities' populations and march them away from their homelands. Sometimes that even included forced labor. However, this was probably more like the earlier Assyrian and Babylonian practice of deportations. Especially rebellious populations would be forced to relocate somewhere where they could benefit the empire, but not have access to familiar allies and resources. Many Greeks were deported as far off as southern Iran or Bactria in modern Afghanistan.

Persian subjects could also be called on to perform labor for the government as a sort of corvee system, where manual labor or military service was effectively part of the tax burden. This summons was only ever temporary. At the same time, the Persian kings called all of their subjects bandaka. The word literally means "bondsmen," which could sound like a synonym for slave until you realize that it was used just as much for high ranking nobles as it was for deported peasants.