The piano, besides being a tangible object and plot point, is a symbol in the book. Bankruptcy forces the Sedleys to auction off all their things, including this little piano. Dobbin buys it and sends it back to Amelia. She thinks it's a present from George and it becomes her favorite thing ever. When she finally realizes that Dobbin bought back the piano, she immediately declares that "It was valueless now [...] it was shockingly out of tune" (59.30).
The symbolism isn't hard to tease out, right? The piano is a neat stand-in for Amelia's feelings for the two men and their feelings for her. She is deluded enough to think that George would buy it for her; selfish George would never think of doing such a thing; and Dobbin is so meek and pathetic that not only does he buy her the piano, he doesn't even correct her when she thinks George bought it. Seriously, get a spine, dude.