Let’s look at the quote itself first.
“Sir,” said Grant, “I have no doubt in the world that the sole object is the restoration of the Union. I will say further, though, that I am a Democrat—every man in my regiment is a Democrat—and whenever I shall be convinced that this war has for its object anything else than what I have mentioned, or that the Government designs using its soldiers to execute the purposes of the abolitionists, I pledge you my honor as a man and a soldier that I will not only resign my commission, but will carry my sword to the other side, and cast my lot with that people.”
First, was Grant a Democrat in 1861? Well, he certainly leaned in that direction. He had voted for James Buchanan in 1856, although, as he explained, it was more a vote against Republican candidate John C. Fremont than a vote for Buchanan. Later he expressed his support for Stephen A. Douglas, although he did not seem adverse to drilling the Wide Awakes, who marched in support of Abraham Lincoln’s candidacy in 1860. Although he did not vote in that election, he had befriended Galena’s leading Democrat, John A. Rawlins, and it had been Rawlins’s speech after the firing upon Fort Sumter that helped move Grant toward reentering military service. True, by this time Grant had also encountered Republican congressman Elihu B. Washburne, but no one identified Grant as a Republican. If anything, he was a War Democrat, although he wore his party allegiances lightly: he was most likely partial to the Whig party in the 1840s and early 1850s.
As for Grant’s regiment, the 21st Illinois, it was recruited from Illinois’ 7th Congressional District, which included Charleston, where in 1858 Abraham Lincoln had bowed to local prejudice and uttered words about his views on race that would forever follow him. In 1860 Democrat James C. Robinson became the third member of his party elected to represent that district, whose boundaries had been drawn in the wake of the 1850 census, in which Illinois had picked up two seats. So it is likely that a majority of Grant’s soldiers were in fact Democrats.
These two pieces of evidence suggest that perhaps the quote is not so far off as one might suspect … what else would we want to know?
Crossroads 
Perhaps I’m jumping the gun, but I’d want to know if a verifiably authentic source had Grant either claiming or denying he was a Democrat. Such as, perhaps, in his September, 1859 letter to his father in which he said, “You may judge from the result of the action of the County Commissioners that I am strongly identified with the Democratic Party! Such is not the case. I never voted an out and out Democratic ticket in my life. I voted for Buch. for President to defeat Freemont but not because he was my first choice. In all other elections I have universally selected the candidates that in my estimation, were the best fitted for the different offices and it never happens that such men are all arrayed on one side. The strongest friend I had in the Board of Comrs. Is a F. S. but opposition between parties is so strong that he would not vote for any one, no matter how friendly, unless at least one of his own party would go with him. The F.S. party felt themselves bound to provide for one of their own party who was defeated for the office of County engineer; a Dutchman who came West as an Assistant Surveyor upon the publick lands and who has held an office ever since.”
It’s clear Grant was not a Republican, and he himself later said he was fond of Stephen Douglas. I would not identify him as a hardcore Democrat, but in 1860 he was surely not a Republican and had set aside his flirtation with the Know Nothings.
Of course, he would be at some pains to deny to his antislavery father that he was a Democrat, especially given his current dependence on his father for assistance.
Can we learn anything about the editor of the Randolph Citizen, the alleged source?
We have. All will be revealed. 🙂