Why did the acronym GOP stick to the Republicans and not to the Democrats?

One reason the name has continued to stick to present-day Republicans is that the "Grand Old Party" really is Old - both "old" as in demographic, and "old" as in conservative, "old-fashioned values".

Another is that Democrats are less prone to the kind of stentorian self-celebration, and sentimental self-delusion, required to unironically refer to their particular flavor of neoliberalism as "Grand" in any meaningful sense.

Although, according to this article by Christopher Klein, the term 'grand old party' was used to refer to the Democratic party in 1859 by Democratic Governor Beriah Magoffin, and later, in 1860, by a Democratic newspaper, the 'GOP' acronym only appears to have been applied on a large scale to the Republican party.

The original term which associated 'GOP' with the Republican Party appears to have actually been 'gallant old party'. An example of this use was on the Senate floor in 1876 by John A. Logan, a Union general in the American Civil War, and later Republican Senator for Illinois. According to a biography of Logan entitled Life and services of Gen. John A. Logan, as soldier and statesman, during a clash with former Confederate generals in the Senate, Logan used the phrase 'gallant' to characterise the actions of his party during the Civil War.

Sir, we have been told that this old craft is rapidly going to pieces; that the angry waves of dissension in the land are lashing against her sides. We are told that she is sinking, sinking, sinking to the bottom of the political ocean. Is that true? Is it true that this gallant old party, that this gallant old ship that has sailed through troubled seas before, is going to be stranded now upon the rock of fury that has been set up by a clamor in this Chamber and a few newspapers in the country?

This appears to then have been conflated with the 'Grand Old Party' moniker - papers from around the time use the term to refer to the party in the same context - referring to the party's work to preserve the Union in the Civil War. This terminology was then used significantly in the press, and in debate, while the Civil War was still in the recent past - helping to cement it as specific to the Republican Party.

It seems, therefore, that rather than referring to the age of the party, it was originally a reference to the party's actions. As to why the acronym stuck - Klein's article suggests that it was due to a typesetting issue:

“Safire’s Political Dictionary” reports that the Republican’s GOP acronym began to appear in print in 1884. Newspapers in 1936 credited T.B. Dowden, a Cincinnati Gazette typesetter, with coining the initials after receiving a story about 1884 Republican presidential nominee James Blaine shortly before press time that ran too long. “My copy ends with ‘Grand Old Party,’ and I have two words left over after I’ve set the 10 lines. What shall I do?” Dowden asked his foreman. “Abbreviate ’em, use initials, do anything, but hurry up!” came the reply. In a rush, Dowden shortened the name of Blaine’s planned speech from “Achievements of the Grand Old Party” to “Achievements of the GOP.”