Originally Posted by
Often1
The US now deals with this by requiring both parents' signatures on the child's passport application or proof that the custodial parent is the sole custodian or living parent.
Not sure how it's done in Europe, but this takes away the guesswork.
The above comment is about (US) passport acquisition for a minor child, but the OP is asking about entry requirements to travel with her own minor child who has a different family name -- that is a separate issue from passport acquisition requirements.
The US passport acquisition requirements are irrelevant to the situation about which the OP is inquiring.