Origin of pluralisation of verbs and nouns in English

From this question, I was just wondering why plural nouns use the ending -s, while the exact same ending is used for the third person singular form of verbs.

How did we get into this weird situation? Why don't we use the same plural ending for both nouns and verbs?

The English plural -s is the only survivor of a much more complicated Old English nominal declension system. As you can see from the tables on the linked Wikipedia article, the plural ending for the Nominative and Accusative of "strong masculine nouns" was -as, and as the Old English nominal system broke down, this ending was generalized to all nouns in all cases. By Middle English we only have the ending -es for all nouns, and in Modern English the -e- has disappeared (except in spelling in some cases), giving us the plural -s.

The verbal ending -s for the third-person singular in the present tense comes from someplace completely different. As late as Early Modern English (the King James Bible, Shakespeare, etc.) the 3sg ending was -eth, which in turn goes back to an Old English ending -eþ (though as above, the full Old English system was far more complex than anything in Middle English or Modern English). A sound changed turned the ending -eth into -es, and the vowel was lost by the same process that eliminated the vowel from -es in the plural.

Thus it's mostly a perverse coincidence that the ending for plural nouns is also used with singular verbs. The two endings have completely different origins, and now look alike because sound changes turned both of them into the same thing.

3rd person singular 's' does not derive as a phonological modification from Early Modern English or Middle English 'eth', but from Old Norse, the dialects spoken by Scandinavian invaders/settlers whose language merged with Old English in the Danelaw and produced major lexical, grammatical and phonological change.*

This merging process took place between 900 and 1200 AD, by which latter date the Danelaw and separate Anglo-Danish identity had long disappeared. The process took a dual form: in some areas all persons uniformly acquired final 's' to the root of the verb and, in others, the several OE endings simply disappeared, resulting in so-called zero endings.

Gradual modification and standardisation, from the 17th century onwards, produced the modern 3rd person singular of the simple present tense and zero endings for all other persons. The entire dual forms still exist for all persons in non-standard English ('was' and 'were' are also examples) with varying levels of 'prestige' i.e: social acceptability, but, like bad girls, they've gone everywhere. They are habitual speech forms wherever English is spoken by native speakers.

Sez you? I hear you cry. Well, I guess it don't matter where because you pays your money and you takes your choice.

*A phonological modification of final 's' to final 'r' occurred in Old to Middle Norse (eg: Old Icelandic) during this period, but did so in Scandinavia and Iceland and seems to have been independent of the ongoing linguistic situation in Britain. Some experts maintain that the 's' ending for all persons possibly originated in a supposed Scandinavian difficulty in pronouncing the 'th' endings of the OE verb paradigm. I consider this unlikely because ON had both voiced and voiceless 'th' endings, as did all other Germanic languages of the period.