Are strings and branes made of underlying quantum fields, just like particles in QFT. Or are they really the fundamental entities of nature and not made out of fields?
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Strings are treated as the fundamental objects in that we obtain the theory (at least in certain limits) by quantizing the dynamics of a relativistic string. However, the precise language used to describe this quantization is just ordinary quantum field theory, but we interpret it differently. Usually, the fields live in spacetime and take values in some abstract internal space (like the fibers of a vector bundle over spacetime). In the usual approach to string theory, the quantum fields live on the trajectory of the string (the worldsheet) and take values in physical spacetime. This is no different from reading quantum mechanics as a 1 dimensional field theory living on just the time direction (trajectory of the particle), we just now have this interpretation for two dimensional field theories as well.
However, there is another approach (string field theory) where the strings are seen to be created from string fields, which is more close to the usual QFT story. However, strings, as one dimensional objects, have infinitely many more degrees of freedom than point particles, so the mathematical heavy lifting of string field theory becomes much more cumbersome and it is typically inconvenient to talk about string theory this way.
Also, a lot is learned about string theories by studying their low energy effective dynamics, which is typically captured by some sort of QFT action. So the answer is really that yes, strings are viewed as new fundamental objects in their own right, but we describe them using the same old tools as everything else (quantum fields) because it’s the best way we know how.