Maschine is a hyperonym of Flugzeug: the latter is a kind of machine, but there are many other kinds of machines.
Seine Maschine flog nach Berlin.
In a sentence like this, the context makes it pretty clear that Maschine refers to an airplane. This figure of speech is called synecdoche. There are different types of synecdoche.
- Er ging nach Amerika. (=die Vereinigten Staaten)
totum pro parte (the whole for the part)
- Das kostet fünf Euro pro Kopf. (=Person)
pars pro toto (the part for the whole)
- Hast du ein Tempo? (=Taschentuch)
species pro genere (the species for the genus)
- Wir Zweibeiner glauben, daß wir die mächtigsten Wesen dieser Erde sind. (=Menschen)
genus pro specie (the genus for the species)
Maschine for Flugzeug is of the last type; the genus is used for the species.
These figures of speech can become lexicalised, i.e. conventional. In fact, Duden has Flugzeug as meaning 2a) of Maschine. The same holds for all the other examples I gave: Person is listed in dictionaries as one meaning of Kopf etc.