I was following the question What is a D-brane? and I want to ask further: Does each open string have its own D-brane or pair of branes? Do elementary particles arise from oscillation of strings onto their own D-branes? Is there an object in QFT that is equivalent to a D-brane?
Before the "superstring revolutions" during which it was realized that one should view branes not only as Dirichlet boundary conditions for open strings but as dynamical objects in their own right, the only significance of D-branes was that they were the subspaces of ten-dimensional space on which the endpoints of open strings moved. So, yes, to every open string there is an associated D-branes or pair of D-branes.
"Oscillations onto their D-branes" is a term without a technical meaning. I sketch in this answer of mine how particles states arise from string theory. The states of open strings are different from those of closed strings, but in the quantum theory, "oscillations" are a heuristic term and the "modes of oscillation" are the creation/annihilation operators for particles just like in QFT. To receive a better answer you'd need to phrase this question more precisely in technical terms.
QFTs have no equivalent object to D-branes - string theory is more than QFT, and in QFT there are no extended dynamical objects that would have boundary conditions defining analogues to D-branes. The closest analogue might be domain walls, which are a rather subtle and large topic in themselves.
Each string has as endpoint a $Dp$-brane. $D$-branes can be coincident, that means one "living" on top of each other. There is no restriction on the number of D-branes a string can have as boundary. If a string is attached to $n$ D-branes on one side and $m$ on the other then we have a $U(n) \times U(m)$ system that can be interpreted as a gage theory with fields living transforming in the bifundamental representation say.
I a do not know exactly how "particles", as viewed by pheno people, originally arise but D-banes by themselves give rise to gauge theories. E.g. $n$ coincident D3 branes give rise to a conformal gauge theory with gauge group $U(n)$. In the context of brane engineering, attaching other types of branes to the system (in specific ways) can give rise to matter representations as well. Brane engineering has its origins in the Hanany-Witten constructions (check their famous paper). Ibanez's and Uranga's book discuss a lot of string phenomenology and how to obtain SM like field theories.