The opposite approaches to make plural form for nouns and verbs

Solution 1:

Yes, it does seem related, but in fact it is not. If you take a look at etymonline, it will tell you that:

-s (2) is third person singular present indicative suffix of verbs, it represents Old English -es, -as, which began to replace -eð in Northumbrian 10c., and gradually spread south until by Shakespeare's time it had emerged from colloquialism and -eth began to be limited to more dignified speeches.

Whereas:

-s (1) suffix forming almost all Modern English plural forms of nouns, gradually extended in Middle English as -es from Old English -as, the nominative plural and accusative plural ending of certain "strong" masculine nouns (such as dæg "day," nominative/accusative plural dagas "days"). The commonest Germanic declension, traceable back to the original PIE inflection system, it is also the source of the Dutch -s plurals and (by rhotacism) Scandinavian -r plurals (such as Swedish dagar).

The same link adds the interesting mention that:

The triumphs of -'s possessives and -s plurals represent common patterns in language: using only a handful of suffixes to do many jobs (such as -ing), and the most common variant squeezing out the competition. To further muddy the waters, it's been extended in slang since 1936 to singulars (such as ducks, sweets, babes) as an affectionate or diminutive suffix. So although there may be "common patterns", they are not etymologically linked.

Thought.co sheds more light on the matter:

The Evolution of English: From -eth to -(e)s

"The Renaissance brought several changes in English grammar and syntax. In the sixteenth and seventeenth centuries, the –eth third-person singular verb ending (e.g., followeth, thinketh) began to die out, though some common contractions of these forms (e.g., hath for haveth, doth for doeth) persisted into the late seventeenth century." (The Broadview Anthology of British Literature, 2nd ed., ed. by Joseph Black, et al. Broadview Press, 2011)

"[W]e know that the originally northern third-person singular verb ending -(e)s spread conclusively to the south during the early modern English period to give she walks, he writes. Nevertheless, there is an ostensibly odd, opposing development whereby some Scots writers at this time adopted the otherwise declining southern -(e)th (e.g. she helpeth), retaining it right into the seventeenth century. A closer examination of the corpus data shows that many of the verbs with -(e)th, in fact, have a stem ending in a sibilant sound, like ariseth, causeth, increaseth, produceth." (April McMahon, "Restructuring Renaissance English." The Oxford History of English, rev. ed., edited by Lynda Mugglestone. Oxford University Press, 2012)

About the plural in nouns this source says

Thanks to the Germanic invaders and Scandinavian Vikings in the 8-11th centuries AD, English became much more simplified. Originally the rules of plurality varied greatly: multiple goat became gak, lamb became lambru, house was house, eye was eyen, and others simply added the plural –s as in day becoming days. The many differences confused the invaders. The invaders oversimplified the Old English by simply adding –s to a singular noun to make it plural (with a handful of exceptions, still). Eventually, the number of such invaders grew, many married natives of England, and thus the “new” English and the plural –s began... just as the Scandinavian invaders influenced it, German, Dutch, Latin and Ancient Greek, as well as many other modern languages have influenced Modern English.

So, while in verbs -(e)s comes from the Northumbrian -(e)th, in nouns it comes rather from invaders (most probably Scandinavian), who wanted to simplify the language.

Solution 2:

As @fev has pointed out, etymologically they don't appear to come from the same sources. But they have arrived in the same place and they've been there a very long time, so that a lot of speakers have identified them as the same, whatever their origins.

Consider the fact that all three {-Z} morphemes: hammers (3sgpres), hammers (pl), hammer's (possessive) have identical allomorphs. All of those hammer words end in /z/. But bolts, bolts, and bolt's all end in /s/, and mashes, mashes, and mash's all end in a /z/ with an epenthetic schwa, providing an extra syllable. This is not an accident.

It is, in fact, what you find when an inflectional system is decaying; English's system (notice that extra syllable in English's?) is practically gone. Of a maximum of 9 inflectional morphemes, 3 are identical. We don't really have much use for them any more; they're pretty much decorative now.