Yes, but there's effectively a hierarchy, just as there is in English. A pushbike is a "Fahrzeug" because it's a thing you can travel on, but more specifically it's a "Fahrrad" or indeed just "Rad" for short. I've never heard of one described as a "Wagen", that seems to require more wheels.
Of course without doubt the best vehicle in Germany is the "Luftkissenfahrzeug" - literally "air cushion travel thing" (hovercraft). But as that isn't wheeled, it's not a "Wagen". Whereas a car is both a Wagen (a thing that moves on wheels) and a Fahrzeug (something you can use to travel). And you can prepend "Kraft" to indicate that it's self powered i.e. has an engine, i.e. Kraftwagen/Kraftfahrzeug. (Though a DMU or EMU is a "Triebfahrzeug", which is again a bit different - and can also be a "Triebwagen", though I'd be inclined to say that a 2-car DMU is one "Triebfahrzeug" made up of two "Triebwagen").
And we said English was weird.