Yes, the excerpt is from his memoirs. He didn't have his dress uniform with him since it was in a long-lost baggage wagon far behind Grant's location at that moment. The past seven days had all been a mad rush to try to outrun Lee, and Grant's baggage was who knows where along that miles-long line.
The story is indeed very moving. Here's another moving story that's sure to touch your heart, this one
about Grant,
by James "Pete" Longstreet, Lee's right-hand man and also formerly a dear friend of Grant from "the old army" -- he had even been at Grant's wedding in St. Louis back in 1848.
"Years later [in the late 1850s, after Grant had left the army], I happened to be in St. Louis and there met Captain Holloway and some other army chums. We went into the Planters House to talk over old times, and it was soon proposed to have an old time game of brag [a version of poker], but it was found we were one short of making up a full hand. Then a poorly dressed man in citizens clothes came in and whom we recognized as our old chum Grant. Going into civil life Grant had been unfortunate, and he was really in needy circumstances.
The next day I was walking in front of the Planters when I found myself face to face with Grant, who, placing into the palm of my hand a 5 dollar gold piece, insisted that I should accept payment of a debt of honor 15 years old. I peremptorily declined to take it, alleging that he was out of service and more in need than I. 'You must take it,' said he. 'I cannot live with anything in my possession that is not mine.' Seeing the determination in the man's face, and in order to save him the mortification, I took the money and shaking hands, we parted. The next time we met was at Appomattox, and the first thing that General Grant said to me when we stepped inside, placing his hand in mine was, 'Pete, let us have another game of brag, to recall the days that were so pleasant.' Great God! I thought to myself, how my heart swells out out to such magnanimous touch of humanity. Why do men fight who were born to be brothers?"