familiär
familial
Relating to or belonging to the family.
Aus familiären Gründen muss ich die Reise leider absagen.
Unfortunately, I have to cancel the trip for family reasons.

familial / familiar
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familial
Relating to or belonging to the family.
Aus familiären Gründen muss ich die Reise leider absagen.
Unfortunately, I have to cancel the trip for family reasons.

The adjective 'familiär' (familial/familiar) is derived directly from 'die Familie' (family). It describes things that have a connection to the family or show characteristics of a family.
Das Restaurant ist für seine gute Küche und den familiären Service bekannt.
The restaurant is known for its good cuisine and friendly, family-like service.
In this meaning, 'familiär' (familial) refers to the structure and affairs of the family. Typical combinations are 'familiäre Gründe' (family reasons), 'ein familiärer Hintergrund' (a family background), or 'familiäre Verpflichtungen' (family obligations).
Er hat eine starke familiäre Bindung zu seinen Großeltern.
He has a strong family bond with his grandparents.
This meaning describes an atmosphere, not the family itself. When people speak of a 'familiäre Atmosphäre' (family-like atmosphere) or 'familiärer Umgang' (friendly, family-like interaction), it means that the mood is very personal, relaxed, and warm, often also in the workplace.
Trotz der Größe des Unternehmens herrscht hier eine sehr familiäre Atmosphäre.
Despite the size of the company, there is a very family-like atmosphere here.
This meaning is used almost exclusively in medicine or biology. Signal words in the sentence are terms such as 'Häufung' (clustering), 'Belastung' (burden/history), or 'Veranlagung' (predisposition) in relation to diseases.
Bei diesem Syndrom gibt es eine bekannte familiäre Veranlagung.
There is a known familial predisposition with this syndrome.
personal / in person
'persönlich' and 'familiär' can both express closeness between people. 'persönlich' fits when something is individual or directly concerns a specific person. It can be neutral, for example with personal questions or personal experiences. 'familiär' describes more of a warm, relaxed, and friendly atmosphere, almost like in a family. The words can be confused when a situation feels close and friendly. The helpful question is: Is it about an individual person? Then 'persönlich' is more precise. Is it about the mood or atmosphere? Then 'familiär' is more natural.

In dem kleinen Hotel war die Atmosphäre sehr familiär.
In the small hotel, the atmosphere was very friendly and family-like.
Im Gespräch stellte die Chefin einige persönliche Fragen.
During the conversation, the boss asked a few personal questions.