Welcome to the forum, High on grammar.
A ship's doctor is resident on board and his full-time job is to care for the crew and passengers.
No school is big enough or wealthy enough to employ its own doctor, so a school doctor will visit a number of schools.
(We do not have school doctors in the UK.)
The articles 'a' or 'the' before either of these terms do not make them English or non-English.
'He is a ship's doctor' means he works on an unspecified ship.
'He is the ship's doctor' is how the captain of a particular ship would refer to him.
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