Passport validity requirements for a EU resident (but not a EU citizen) to travel within Schengen

Even if there are general rules on which documents are accepted when entering or leaving the Schengen area, it is up to each member state to decide which documents are recognized and required for a foreigner to stay in the country. This has the odd implication that you as a citizen of a Schengen state don't need a passport to cross the border to another member state, but may be required by national law in the other state to be in posession of a passport to stay there.

For Germany, you can find the list of officially recognized foreign travel documents here (in German). There is no absolute requirement that passports may not be valid for more than 10 years, there are however for several countries restrictions that only passports issued after a said date are valid. One odd example would be Turkmen passports, which are valid when issued after 1996, even if they have no expiration date at all.

As you can see from the German regulations (16 pages of fine print), it is not possible to give you a general answer. As Mark Mayo suggested, if you ask if a specific document is valid, you may get better help.

Ukrainian regular passports are in Germany valid without any restrictions. The symbols and abbreviations used in the German list are not quite intuitive to understand, but "JA 3) (2000)" means that the 2000 series of the passport is recognized, but hat earlier issued passport are also still recognized as long as they are not expired. Remark 3) is not applicable to regular passports.