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As noted in the comments, under the Schengen Borders Code there is generally no border control between Schengen countries (Article 22), but police may perform "spot checks" nevertheless, in order "to combat cross-border crime" (Article 23).
Furthermore, countries may temporarily reintroduce border controls on the internal borders (Chapter II, Articles 25 through 35). I do not know whether that is currently the case between France and Germany, but even if that is the case it has no bearing on answering your question.
This is because crossing an internal border, does not constitute entry into the Schengen area, regardless of whether the traveler's documents are checked. Rules governing entry into the Schengen area do not apply. In particular, the rule concerning the validity of the traveler's passport may be found in Article 6, which is part of Chapter I, Crossing of external borders and conditions for entry. The rule is in paragraph (1)(A)(i):
- For intended stays on the territory of the Member States of a duration of no more than 90 days in any 180-day period, which entails considering the 180-day period preceding each day of stay, the entry conditions for third-country nationals shall be the following:
(a) they are in possession of a valid travel document entitling the holder to cross the border satisfying the following criteria:
(i) its validity shall extend at least three months after the intended date of departure from the territory of the Member States. [...]
(As an aside, note that the rule isn't actually that the passport must be valid for six months beyond your date of entry, but that it must be valid for three months beyond your anticipated date of exit.)
Since this rule is part of the conditions for entry into the Schengen area, it does not apply to those who are crossing internal borders. If you tend to worry about these things, though, there's no harm in getting a new passport now.