Is the mechanism of solar flares on red dwarfs and brown dwarfs the same as that on the Sun?

Yes, the basic mechanism is thought to be the same on red dwarfs and at least the hotter brown dwarfs, but the details can be different.

As you say, magnetic reconnection in the corona is the starting point. Well, actually it is fluid motions at the magnetic loop footpoints that is the starting point. The B-field and partially ionised plasma are coupled and photospheric motions put (magnetic) potential energy into the B-field structures.

This potential energy can be released suddenly in reconnection events. These can drive coronal mass ejections or accelerate charged particles along the loop field lines.

A flare occurs when a significant amount of energy goes into accelerating charged particles down the field lines towards the loop footpoints. These charge particles emit radio waves and then non-thermal hard X-rays as they impact the thicker chromosphere/photosphere. Their energy is then thermalised, heating the chromosphere and possibly causing hot ($>10^{6}$ K) material to evaporate into the coronal loops. Here it can cool by radiation and conduction before falling back, or perhaps forming cool prominences.

Similar things must be going on in low-mass stars. The X-ray and optical light curves of their flares do resemble what is seen on the Sun as do the relationships between hard- and soft X-rays and the evolution of plasma temperatures. The details may be different because the temperature and density structure of their photospheres, chromospheres and coronae are a bit different to the Sun's, and there are some indications that their coronae can be denser or sometimes that flares occur in much larger structures than are seen on the Sun. "White light" flares are also more common in M-dwarfs.

Why are flares on red dwarfs so powerful? Partly it is contrast - you are comparing the flare emission with something that is intrinsically less luminous. In absolute terms the flares on the Sun and flares on red dwarfs are not hugely different. What is different is that the flare power as a fraction of the bolometric luminosity and the frequency of large flares can be much higher on M-dwarfs.

The underlying reasons are likely to do with the magnetic field strengths and structures on red dwarfs. Magnetic activity is empirically connected to rotation and convection. Magnetic activity is higher on rapidly rotating stars and those with deep convection zones. M-dwarfs have very deep convection zones (or are even fully convective). They also tend to rotate much more rapidly than the Sun, since their spin-down timescales appear to be much longer than G and K dwarfs. They are thus more magnetically active relative to their bolometric luminosities. It appears that active M dwarfs have very strong magnetic fields (like those in sunspots) covering a very large fraction of their surfaces and this along with the convective turbulence where the magnetic loop footpoints are anchored likely leads to their strong flaring activity.

Brown dwarfs are a bit more tricky. The younger, hotter ones probably behave much like low-mass M-dwarfs (in fact they are M-type objects). The magnetic activity on cooler/older L- and T-dwarfs is much more mysterious. I guess a few flares have been seen, but it is not clear this is related to similar mechanisms as in higher mass and hotter stars. These cool brown dwarfs have neutral atmospheres and the magnetic field is not frozen into the plasma like it is in the partially ionised photospheres of hotter stars. This means that the loop footpoints may not be stressed by photospheric motions in the same way. It is not even clear that brown dwarfs generate a magnetic field in the same way as more massive stars, though it is clear that at least some of them do have magnetic fields.