Is there an explanation for why both "pop" and "pops" are used to mean father (singular)?

According to the OED, the s in pops is:

A shortened form of the hypocoristic dim. suffix -sy, added to the same classes of words, as Babs, Toots, ducks, moms.

Also, the earliest citation in the OED for pops is from 1893:

Yes, Libby; while you have been galivanting around Europe gettin' your edication, your old Pops has been peggin' away until he has grown rich as mud.
_{Puck (N.Y.)}

(In comparison, according to this answer, toots is first attested in 1891.)

"Pops" seems to have originated as a variant form of "pop", which is itself a clipped and altered form of the longer "papa". In English, a number of informal clippings have an "s" suffixed to them; I'm not sure exactly why, but this -s does not indicate that they are plural nouns.

Examples:

• totes < totally (discussed in this Language Log post: Totes, where commenters also mention adorbs < adorable).

• preggers < pregnant and turps < turpentine (mentioned in "Clipping from the word-formation, word-class, stylistic/register, semantic and translational perspectives", by Richard Skala, 2006)

The Oxford English Dictionary entry for pops indicates that it is formed from the clipping pop combined with a suffix -s that it says is

A shortened form of the hypocoristic dim. suffix -sy suffix, added to the same classes of words, as Babs, Toots; ducks (see duck n.1 3c), moms.

I wasn't familiar with the referenced singular use of moms, but the OED entry for that word provides some examples that are similar to the use of pops:

In quot. 1976, addressed ironically to a man.

1925 W. Faulkner Let. 20 Feb. in Thinking of Home (1992) 185 What's the trouble, moms? I know something is wrong from your last letter.
1946 M. Mezzrow & B. Wolfe Really Blues 47 Moms always used to make sugar sandwiches for me when I was a kid.
1976 N. Thornburg Cutter & Bone xi. 260 ‘Now, don't you fret, moms,’ he said.
1992 Premiere Jan. 101/1 Because of him and my Moms, Jacquelyn, I was introduced to the arts at a very young age.

Author William Safire, in a New York Times article, suggests that the use of the final “s” in terms like pop and mom probably adds a connotation of “detached intimacy” to those familiar terms:

What has become of Mommy and Daddy, and Mom and Dad?'' asks Isabelle Bradley, an irate mother from Birdsboro, Pa. ''We are becoing a nation of 'Moms' and 'Pops.'''

I have been wondering for some time why my teen-age children have been calling me ''Pops, '' which is the name of an orchestra in Boston. My suspicion is that the addition of an ''s'' to either Mom or Pop connotes an irreverent affection, or detached intimacy, that enables a young person to address a parent in a manner familial yet cool.

''Mama'' and ''Papa,'' the endearments of earlier generations, are becoming rare; they recall immigrant days, or our parents' recollections of their parents. ''Ma'' is used occasionally, ''Pa'' rarely. Like ''Dad,'' from ''dada,'' ''papa'' is rooted in baby talk, and led to an 1838 citation for ''pop,'' meaning ''father.'' Nowadays, ''Pop,'' when capitalized and used as the name for someone who is not your father, refers to a likable geezer or the grizzled night watchman.

The declension of filial address begins with a breezy ''Howzit goin', Pops?'' to a more respectful ''About my allowance, Dad,'' to a resigned and resentful ''Yes, Father.'' Going back up the scale, it is ''Thank you, Mother, I can do it myself'' to ''Mom - can you come over and baby-sit?'' to ''Seeya, Moms!''

If you are disconcerted by the addition of ''s'' to Mom and Pop, try answering by calling your son ''Sons'' or your daughter ''Daughters.'' If they seem puzzled, tell them it's the new middleaged lingo.