The first bunch are indeed "a hidden regularity", just like a fossilized skeleton. The second bunch are a different phenomenon completely that I won't touch on here.
The first bunch are all evidence of what Indo-Europeanists call the "Yodated Causative", a -y suffix that formed a causative/inchoative stem when added to a verb root, just like a later en causative/inchoative affix did: black ~ blacken, white ~ whiten, red ~ redden, joy ~ enjoy, rapture ~ enrapture, light ~ lighten ~ enlighten, etc.
The yodated causative was productive in Indo-European, and many verbs formed common causative/inchoative stems with it. After it stopped being productive, the stems it left behind gradually got reinterpreted as separate verbs.
But most of those verbs still share one physical feature: final Palatalization. A high front semivowel like IPA
[j] (called Yod in I-E studies, from the Hebrew letter י; it's the first phoneme in yellow) is actually a very short unstressed high front vowel [i], and high front vowels (and semivowels) tend to be anticipated by speakers by moving the tongue toward the palatal area, producing a palatalized consonant out of whatever comes before the
In modern English, that means they end in a palatal consonant or cluster, like /ʃ,ʒ, tʃ/, or /dʒ/, which is true of the second verb in each pair of verbs in the first bunch. They all end in a palatal, and they all mean -- or once meant, to be more precise -- 'to come to
V' or 'to cause
V is the first verb in each pair. There are some vowel changes, often cases of umlaut, which also anticipates palatal vowels.
- cling, clench (see other kl- words for a rather different hidden regularity)
- hang, hinge
- meng, minge (now dialectal)
- stink, stench
- drink, drench ("You can lead a horse to water, but you can't drench him")
- tint, tinge
- wring, wrench
All the pairs in the first bunch have nasals in their endings; this is not necessary and there are lots of pairs that don't contain an /n/. For instance
Indeed, yodated causatives are the first thing one should suspect when encountering an English verb ending in a palatal consonant; if you look around, you may well find its noncausative counterpart, too. Two for the price of one.
As to the second group, the intrusive -n- of message ~ messenger etc is treated in the OED in the entry for passenger:
In late ME. n was phonetically inserted before -ger /-dʒər/ as in some other words, including harbinger, messenger, ostringer, porringer, scavenger, wharfinger, etc.: cf. also popinjay.
(See Jespersen in Engl. Studien XXXI. 239.)