What mother disowns her daughter over the latter’s choice of club activities? Probably the kind that takes care to remind her daughter of the ‘proper distance’ between them with every sentence she speaks.
It is not unique to Japanese that very polite and formal language appears cold and distant from the listener. But with the greater complexity Japanese offers in terms of politeness levels of speech, the words chosen show much more about relationships between people than in many other languages.
Mrs. Isuzu apparently opts to address her daughter by <first name> + <san>. Considering the "first name basis" here to be mostly a necessity (as the two obviously share their last name), the main politeness indicator would be the <san> ending. It is commonly thought of as a (rough) equivalent to the western Mr./Mrs./Miss, which makes it quite formal indeed. Of course, it might seem silly for a western parent to address their child like that, but it is not unheard of among Japanese parents. Sometimes it is a matter of treating the child as an individual, without the condescension inherent in calling a child with a <kun> or <chan> ending. Sometimes the situation is just too formal for anything else. But here, it is just part of the refined image Mrs. Isuzu tries so hard to uphold.
But if it is just about keeping up appearances in public, it might get better when in private, right? It does not seem so, as out of the gazillion ways Japanese has of saying “you”, Mrs. Isuzu decides to stick to <anata>. This particular pronoun can be linked to high-class and good upbringing, but it is as often seen as an easy way out when you do not know (or care about) somebody’s real name. After all, <anata> is as impersonal and neutral as Japanese gets, a word belonging mostly to language textbooks and Internet surveys.
This kind of environment had to leave its mark on Hana. The pressures of her upbringing and the expectations resting upon her shoulders are to be found in every sentence Hana speaks. From ending sentences in <desu wa>, through constantly using every polite noun and verb form you can think off (<hito> –> <kata>, <iru> –> <irassharu>, …) and up to complaining of a stomachache in historical play style (<jibyou no shaku ga>), Hana is walking proof of her mother’s principles.
And of course, her name. Hana – the flower – obviously written in the traditional and more difficult way (花 –> 華), leaves little doubt about what Mrs. Isuzu expects her daughter to do with her life. Hana seems to have adapted to this way of way of life so well so far that the standoff between mother and daughter shown in episode four might have well been their first. And although Hana is perfectly polite throughout the entire exchange, it must have been a shock for her mother that Hana can even have a mind of her own.
In stark contrast to the Isuzu family, the Akiyama household is much laxer in its approach to how they address each other. The head of the family addresses his wife by <omae>, which most people probably know as a rude form of address between males. Here, it actually shows how much at ease Yukari’s parents are with each other, not needing to hide behind any formalities. Mrs. Akiyama, on the other hand, points to her husband using the term <otou-san>, literally “father”. This is a typical leftover from the child raising period where everyone in the family is addressed from the child’s perspective, and it subtly implies a warm family atmosphere. The only highly polite language in those scenes is clearly humorous in nature – the ritualistic greeting from Yukari’s father coupled with a dogeza bow (normally reserved for apologizing for grave offences).
The difference between being polite and distant is best shown through Yukari’s mother. You can hear her using polite noun forms (<o-tomodachi>) coupled with informal verb forms to create a good balance when addressing her daughter’s friends – people she wants to be polite to without making the atmosphere too stiff.
On the topic of family, we also have the slowly unraveling mystery of why Mako reports to her grandma and not her parents (considering her reaction to the Akiyama family photo, it’s likely more than a matter of convenience), early hints pointing to Miho’s harsh military-style upbringing and Saori’s comments about her doting father. All things considered, Girls und Panzer presents a set of characters with widely different familial backgrounds, something of a rarity with so many “disappearing-parents” anime.