The situation in germany is similar to the United States:
Power of Attorney does not extend to voting in a political election.
Some details:
Under German law, the equivalent to power of attorney due to mental health concerns would be a Betreuung (if it applies to all affairs) or a Pflegschaft (only for some areas, such as only for representation in a specific court case).
A Pflegschaft is, as mentioned, only for specific areas where (such as representation in a court case, health decisions, or financial decisions) and simply does not exist for voting in elections.
For a Betreuung, which is typically granted for individuals who have severe mental disabilities or suffer from dementia (and must be ordered by a judge):
Until 2019, someone under a complete Betreuung lost their right to vote. Since this was considered discriminatory towards people with disabilities, this restriction was lifted in 2019. Since then, the law only says (after stipulating that in general a vote must be cast by the voter personally):
(5) Ein Wahlberechtigter, der des Lesens unkundig oder wegen einer
Behinderung an der Abgabe seiner Stimme gehindert ist, kann sich
hierzu der Hilfe einer anderen Person bedienen. Die Hilfeleistung ist
auf technische Hilfe bei der Kundgabe einer vom Wahlberechtigten
selbst getroffenen und geäußerten Wahlentscheidung beschränkt.
Unzulässig ist eine Hilfeleistung, die unter missbräuchlicher
Einflussnahme erfolgt, die selbstbestimmte Willensbildung oder
Entscheidung des Wahlberechtigten ersetzt oder verändert oder wenn ein
Interessenkonflikt der Hilfsperson besteht.
Bundeswahlgesetz - § 14 Ausübung des Wahlrechts
Translation (no guarantees, legal German is hard to translate):
(5) A voter who cannot read or cannot cast a vote due to disability
may use another person's help. The help is limited to technical help
to declare a voting decision made and communicated by the voter
themselves.
Help is not permissible if it uses illegal influence, if it replaces or
changes the decision process or the decision of the voter, or
if there is a conflict of interest for the person helping.
This makes it abundantly clear that only the person eligible to vote may make the actual voting decision.