That's true. Bremsstrahlung is one of the few German words making it into English. Literally translated it means "braking radiation" ;-)). The name comes from the early days of electrons (discovered by J. J. Thomson 1897) and electron theory. It just describes the phenomenon that accelerated charges radiate. It is not specifically quantum but a prediction of classical electromagnetism (Maxwell's greatest discovery). It's described by the solutions of Maxwell's equations with accelerated particles as the four-current source of the electromagnetic field, the retarded potentials (or the electromagnetic field itself, i.e., the Jefimenko equations). This special solution is also known as the Lienard-Wiechert potentials.
Quantum theoretically the only correct description is relativistic quantum field theory as is always true as soon as the electromagnetic field is involved. Often it's said that bremsstrahlung is described by a Feynman diagram, where a photon line is emerging from a charged-particle line (e.g., an electron or positron line in fermionic QED). Ironically this is a prime example for the fact that it can be very misleading, when the "photon explanation" is invoked in a somewhat careless way. It's of course the leading-order diagram to describe, e.g., the emission of em. radiation for an electron moving in a static field of a very heavy particle like an atomic nucleus (which is a good approximation in this case, as can be proven from QED; see Weinberg, QT of Fields, Vol. I). What comes out from this single Feynman diagram is usually the first encounter with an infrared divergence in QFT. The solution is the resummation of soft photon diagrams (including the leading-order radiative corrections for the elastic scattering of the electron on the external field), and this leads more to a field-like picture of the bremsstrahlung than a particle picture in terms of naively understood photons. So the classical-field picture is a good intution for bremsstrahlung even from the point of view of QFT!