I don't know, they sound like the same to me, even when comparing both to the original song.
Well, let's start with the obvious to separate the two as different renditions of a song: at the very beginning of Polygon's video, the song you hear has one guitar playing the main melody and nothing more, while the song playing while trading with Pedro II has, if not two guitars, then one guitar doing the main melody and harmonics.
While that does prove that the two songs are different pieces, it does not prove the two are not renditions of the same original song.
For that, I tried playing the main melody of both on the piano, and see what were the results.
The melody in Polygon's video goes like this:
G E F# D E B D
While the melody in Pedro II's diplomacy screen goes something like this:
A E F# E G# Ex3 F# E F# C# E
Judging from the notes used both melodies, the first is in G Major 7, while the second is in A Major 7, so for a fair comparison, we'll have to transpose one of the songs to the other's main tone - I decided to transpose the first to A Major 7, which becomes like this:
A F# G# E F# C# E
Comparing again with the melody in Pedro II's diplomacy screen:
A E
F# E
G# Ex3 F# E F#
C# E

Yup, they're the same melody, my bad! Sorry about the false alarm guys.
And the same can be done with the original song by Ernesto Nazareth, indeed these are all different arrangements of the same song!
For curiosity's sake, Ernesto Nazareth's Brejeiro's melody goes something like this:
Gx2 Ex2 F#x2 D Ex2 Cx2 Dx2
So it all checks out, the two are indeed extremely similar. For some reason, as the game progresses Pedro's theme raises one step, which threw me off yesterday night (never theorize while tired, the results will be wrong

)
Or maybe I'm just doing this all wrong!

I'm no expert in the matter...