"Innings" in British usage is either singular or plural. It's just one of those words with identical singular and plural forms. It's not the only word ending with an s that's plural; consider (apart from many -ics words like physics and politics) news and (both singular and plural) means, series, species, etc. This is what Fowler's A Dictionary of Modern English Usage (British, 1926) says:
Plural anomalies. See -ICS for the question of whether words in -ics are singular or plural. Plural names of diseases, as mumps, measles, glanders, can be treated as singular or plural; chickenpox & smallpox, originally plural, are now reckoned singular. Innings, corps, & some other words in -s, are singular or plural without change in spelling, but, while corps has -s silent in singular and sounded in plural, an innings & several innings show no distinction, whence arises the colloquial double plural inningses. For the plural of Court Martial & Lord Justice, the number of porridge, & the difference between pence & pennies, see the words.
So it was special enough to invite comment (and unusual enough to invite coinages like "inningses"). The OED doesn't give any special etymology; it just comes from "in", as the "outing" you mentioned comes from "out".
In American usage, "inning" is a back-formation from the plural "innings". According to a random comment on languagehat,
… "innings". This is both the singular and plural form in cricket. It is also frequently seen this way in early baseball. For a time both "inning" and "innings" were seen used as singular, but by the 1870s or so the singular "innings" was uncommon. Nowadays it is unheard of in a baseball context.