I recently got into Neural networks. As much as I have understood, the learning process is based on the change in weights according to stimulus and algorithm used in learning. Does this in any way represent the real learning process of our brain?
Yes and no.
Neural networks that use edge weights simulate synaptic plasticity - a key mechanism in the brain operation. But it isn't the only one.
In practice, cognition is the combination of various mechanisms, some we are still unsure of.
Perhaps the most obvious missing ingredient of weighted networks is the lack of frequency/time domains similar to that affecting the brain. Neurons involve time-dependent operation - for instance, both the action potential and the following refractory period are time-bound. The brain also involves synchrony, e.g. the phase-locked discharges of a neuron set.
There're also issues like different axon lengths that play a part. And many more.
In addition, and possibly above all, the actual topography matters a great deal - how everything is connected.
More advance brain simulations involve many of these extra components - mimicking more accurately a real biological brain.
The approach of the artificial neural networks that you describe and their application is called Connectionism. There are a number of cognitive architectures that have used this approach to explain cognition, such as Leabra.
The question about whether this is really what the brain is doing is another question entirely. Does the brain do back-propagation? This is currently up for debate. Does it contain structures such as LSTMs and other Recurrent Neural Networks? Probably not, as these aren't very efficient uses of memory and neurons. For a further discussion of biological plausibility, I would recommend reading "How to Build a Brain" by Chris Eliasmith.