義父, 義母 and 養父: in-laws, stepparents and an adoptive father
Compare support from a father-in-law, life with a new stepfather, cooking with a mother-in-law, and trust with a stepmother. Read these relationships separately from 養父, an adoptive father.
義父 follows the marriage context
義父 (ぎふ) can mean father-in-law or stepfather. In “義父は結婚後も私を温かく支えてくれた。” (“My father-in-law continued to support me warmly after the marriage.”), the father-in-law continued to support the speaker warmly after the marriage. Keep that in-law relationship with this scene.
In “少年は母の再婚で新しい義父と暮らし始めた。” (“After his mother remarried, the boy began living with his new stepfather.”), the boy began living with his new stepfather after his mother remarried. 母の再婚 explains the new family relationship; it does not turn the stepfather into the father-in-law of the first example.
義母 identifies a mother-in-law or stepmother
義母 (ぎぼ) can mean mother-in-law or stepmother. In “妻の母である義母と一緒に夕食を作った。” (“I cooked dinner with my mother-in-law.”), 妻の母 explicitly identifies the wife's mother. The speaker cooked dinner with that mother-in-law.
In “物語の少年は義母と少しずつ信頼を築いた。” (“The boy in the story gradually built trust with his stepmother.”), the boy in the story gradually built trust with his stepmother. Read the stepmother relationship alongside 少しずつ, which describes the gradual development of trust.
養父 identifies an adoptive father
養父 (ようふ) means adoptive father. In “彼は養父から木工の技術を学んだ。” (“He learned woodworking from his adoptive father.”), he learned woodworking from his adoptive father. The relationship concerns adoption, rather than the in-law or remarriage relationships illustrated by 義父.
In “養父との思い出を日記に書き留めた。” (“I wrote memories of my adoptive father in my diary.”), the speaker wrote memories of the adoptive father in a diary. Keep the learning and diary scenes as described, without supplying adoption procedures or additional claims about biological relationships.
Can both 義父 examples be read as accounts of a biological father?
Show the answer and why
No. One identifies a father-in-law who continued to support the speaker after marriage; the other identifies a stepfather after the boy's mother remarried. The shared spelling does not replace those relationships with a biological father.