I'm aware of several verses saying that God is righteous, but what's the biblical basis to say that God is or is not the only righteous entity in the universe.
Paul quotes David in Romans 3:10-15.
Romans 3:10; NASB
There is none righteous, not even one;
There is none who understands, There is none who seeks for God;
All have turned aside, together they have become useless;
There is none who does good, There is not even one.
The language here is speaking of humans. If your question is about non-human beings, then these statements probably don't apply.
Jesus responded to some praise given him, referring to him as “good teacher:”
Luke 18:19; NASB
Why do you call Me good? No one is good except God alone.
The same statement is recounted in Mark 10:18.
Again, the context of this is probably about humans.
I can't find a place where angels are actually referred to as being “righteous.” It is (or was at one time) possible for angels to sin.
2 Peter 2:4; NASB
For if God did not spare angels when they sinned, but cast them into hell and bcommitted them to pits of darkness, reserved for judgment;
However, since we're not told very much about angels, particularly their creation, purpose, fallibility, and place in eternity, it's hard to say very much with certainty about them.