der Erzeuger
producer
A person, farm, or company that produces or grows goods, especially food.
Wir kaufen unser Gemüse direkt beim Erzeuger auf dem Wochenmarkt.
We buy our vegetables directly from the producer at the weekly market.

producer / biological father
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producer
A person, farm, or company that produces or grows goods, especially food.
Wir kaufen unser Gemüse direkt beim Erzeuger auf dem Wochenmarkt.
We buy our vegetables directly from the producer at the weekly market.

The word 'Erzeuger' (producer / biological father) derives directly from the verb 'erzeugen' (to produce/generate), which means 'herstellen' (to manufacture) or 'hervorbringen' (to bring forth). An 'Erzeuger' is therefore someone who produces something—whether agricultural products or offspring.
Dieses Weingut ist ein bekannter Erzeuger von Qualitätsweinen.
This winery is a well-known producer of quality wines.
You can easily recognize the meaning from the context. If it is about agriculture, food, or goods, it means the producer. In a legal or family context, especially if it sounds detached, it means the biological father.
Viele regionale Erzeuger bieten ihre Waren auf dem Bauernmarkt an.
Many regional producers offer their goods at the farmers' market.
In connection with goods, 'Erzeuger' (producer) is often used in fixed combinations such as 'direkt vom Erzeuger kaufen' (to buy directly from the producer), 'regionale Erzeuger unterstützen' (to support regional producers), or in reference to 'landwirtschaftliche Erzeuger' (agricultural producers).
Der Preis ist niedriger, wenn man die Produkte direkt vom Erzeuger bezieht.
The price is lower if you obtain the products directly from the producer.
As a term for the biological father, 'Erzeuger' (biological father) is a very formal, official term. In everyday speech, it can sound detached or even negative because it emphasizes the purely biological role without an emotional bond.
Er kennt seinen Erzeuger nicht und ist bei Adoptiveltern aufgewachsen.
He does not know his biological father and grew up with adoptive parents.
father
'der Vater' and 'der Erzeuger' can both mean the man from whom a child is biologically descended. However, 'der Vater' is the normal and personal word for the male parent. It can also mean a social and emotional role, that is, the man who is there for a child. 'der Erzeuger', by contrast, sounds official, detached, or even negative when there is no personal relationship. Learners may confuse the words because both can refer to a child's origin. In everyday life, people almost always say 'Vater'; 'Erzeuger' is only used when you deliberately want to emphasize the biological aspect or show distance.

In den Unterlagen steht nur der Name des Erzeugers, weil er nie Kontakt zum Kind hatte.
The documents only list the name of the biological father because he never had any contact with the child.
Ihr Vater holt sie jeden Freitag vom Training ab.
Her father picks her up from practice every Friday.