The developments don't exactly seem to be "coincidental", but as far as I can tell, neither is perfectly regular either.
Lengthening in "open syllables"
The quote itself provides an explanation for why Vater may have developed a long vowel:
wie in fast allen betonten silben, gedehnt.
A sound change took place at some point between Old High German and modern German whereby short vowels were lengthened in certain stressed open syllables. That excluded vowels followed by geminate/long consonants or by heterosyllabic consonant clusters.
Actually, my understanding is that this lengthening sound change is the (indirect) reason for the current convention of writing consonants with doubled letters after short vowel sounds. (The use of doubled letters in modern German spelling does not correspond directly to the etymological or regional use of geminate/long consonant sounds.)
Other examples of short a being lengthened (based on Wiktionary's etymologies):
Lengthening apparently occurred fairly systematically when the consonant following the vowel was one of the "lenis" obstruents b, d, g.
But when the following consonant was something else, there are some complicated details about this sound change.
A possible special status for the consonants T and M
One of the complications that I remembered reading about was that word-final -er possibly caused a preceding consonant to geminate in some words (such as Hammer).
When I tried to find more about this, I found a text that suggests that the consonants T and M may have a special status with regard to lengthening:
In High German [...] lengthening is not complete before all simple consonants. The notable exceptions are present not only before following -er, -el, -em, -en, but also before t, m. Oftentimes, these two environments coincide, in which case lengthening is generally absent before m, but may be present before t [...] If there is no lengthening, t and m were geminated, although this is not apparent in Modern Standard German after the simplification of geminates.
("Arriving at the Goal: Vowel Lengthening in Middle Germanic", 188.8.131.52, p. 97 in Gemination, Lenition, and Vowel Lengthening: On the History of Quantity in Germanic, by Kurt Goblirsch)
Other words now written with a double -tt- or -mm- where short a corresponds to modern short a (based on Wiktionary's etymologies):
So even though the lengthening of short a to long a Vater resembles the regular change described in the previous section, lengthening might not have been regular in this particular environment.
It might be a coincidence, but standard English actually shows a similar lengthening in father with an irregular result (/fɑːðə(r)/) and shortening in mother (/mʌðə(r)/). However, unlike German, English also shows shortening in brother.
I don't know why mutter developed a short vowel. Mutter and Vater both end in -ter from Proto-Germanic *dēr, so I don't see how the different development of the vowels could have been conditioned by what followed them.
The source I cited above doesn't seem to talk about shortening of originally long vowels like this (as opposed to retention of an originally short vowel quality). But mach's answer seems to explain this with some examples.
I found another source that says that Middle High German uo sometimes developed to modern German [ʊ], but shortening is described as a "rather
marginal" outcome, with around 93% of MHG uo corresponding instead to long [uː] (p. 196-197, "Vocalic and consonantal quantity in German:
synchronic and diachronic perspectives", by Emilie CARATINI). Caratini disagrees with the idea that -er, -el, -em, -en affected the length of a preceding vowel (p. 274).
The description of shortening as a sporadic rather than regular change in the context of words like Mutter is supported by Russ 1975, who gives a list of vowel shortenings that he says can be considered "real exceptions" to the usual course of development:
The examples usually cited are: NHG Rache, Schach, Amboß,
ansäss(ig), Rüssel, müssen, Jammer, Waffen. Mutter, lassen, MHG
rāche, schāch, anebōz. ansæze, rüezel. müezen, jāmer, wāfen,
muoter, lāzen. For each of these words individual reasons for the shortening must be sought.
(page 277 in Russ, Charles V.J. (1975) "Studies in the Historical Phonology of German" University of Southampton, PhD Thesis)