Is a reflected wave on a string of the same form as of the incident

You already understand how a string can sustain and propagate sinusoidal waves in both directions. Having a reflector at one end is a boundary condition, one that is compatible with the string's motions. Since the string allows propagation both ways the boundary condition representing the model of the reflecting wall constrains the ratio of the reflected and incident sinusoidal waves.

The boundary condition is a model, if the wall is non-linear then the boundary condition is also non-linear and an incident wave may generate harmonics by that model as function of the incident wave amplitude and you get a reflection that is not a pure sinusoidal.

Even the wave generator is represented by a boundary condition that has usually two parts (1) an outgoing wave of given amplitude and a (2) partially absorbing/reflecting wall. The latter is important if there are waves going towards the generator, e.g., usually from reflections from the other end. (In dc electronics the battery is represented by a perfect voltage source plus a resistor, or in the case of an amplifier connected to an antenna the amplifier is represented by an ideal wave source of fixed amplitude independently prescribed and imposed on the rest of the circuit and its output impedance is that partially absorbing/reflecting "wall" to the waves incoming towards the amplifier.)