A brane is not "by definition" "just a soliton".
What is true is that the M2- and the M5-branes are mostly described by being certain "solitonic" solutions to eleven-dimensional supergravity. "Solitonic" here is meant in the broadest sense (not that there really is a single agreed-upon narrow sense, mind you...) of being a stable solution of the equations of motion (in this case of supergravity) that is "localized" and can interact with other such solutions and still retain its original form. This doesn't make the brane a "wave", but it is a soliton configuration of the fields of 11d supergravity. Note also that 11d SUGRA is merely believed to be the "low-energy" approximation of full M-theory, so whether "soliton" describes what an M-brane is actually "meant to be" is somewhat uncertain.
But a D-brane is not defined as a solitonic object at all, it's defined by the Dirichlet conditions for some open strings. That it corresponds, under some dualities (most straightforwardly reduction of M-theory to type IIa string theory), to such solitonic objects as the M2- and the M5-branes is a highly non-trivial insight. Most D-branes also actually correspond to solitonic solutions of the low-energy 10d SUGRA associated to the respective string theories in which they occur. Also, not all the solutions D-branes correspond to are M-branes, for instance, the M-theory dual of a $D_6$-brane is a Kaluza-Klein monopole/Taub-NUT space. Whether we want to call the Taub-NUT solution a "soliton" I'll leave to the reader to decide.
Furthermore, the claim that a D-brane might constitute a "fundamental object" is likewise doubtful - it is an auxiliary object that exists for the fundamental strings to end on, and is not fundamental in original string theory, although nowadays many do treat D-branes as dynamical objects in their own right. Whether that makes them "fundamental" objects is difficult (and potentially pointless) to say.