It is possible for life to develop on the moon of a tellurian planet, if the moon has an atmosphere, there is liquid water, and sufficient time for it to evolve.
Most persons looking at Earth's Moon will observe most of these life-supporting condition are conspicuously absent. Although sufficient time has passed, but, unfortunately, this by itself isn't enough. Therefore, it seems probable that to have a habitable moon those conditions must be present.
Such a moon must be considerably more massive than our Moon. The atmosphere needs a stronger gravity than the Moon's to prevent the losing atmospheric gases in uncomfortably short periods of time. This makes our habitable moon closer to a planet (in terms of mass, size, surface conditions, etc). So this looks more like a binary planet system than an Earth-Moon system of planet and a moon.
Short answer: to have a habitable moon of a habitable planet, the habitable must itself be more like a habitable planet.
As for biological similarity between the lifeforms on the habitable planet and its habitable moon. There will be a high probability of an exchange of meteors between the two bodies. Microbial life has an excellent chance of passing from planet to moon and from moon to planet.
Lifeforms on both worlds will most likely share a common DNA, similar biochemistry, and microbial organisms that are similar. However, evolutionary conditions will shape the majority of lifeforms on either world. While convergent evolution will undoubtedly produce equivalent morphologies or body forms, most lifeforms will be adapted to their own environments.
Expect some lifeforms in common, at least, in terms of appearance and structure, but many will be shaped by the specific conditions of their environments. Chance and natural selection will guarantee biological success. This doesn't ensure the two worlds will have common lifeforms except at the level of their biochemistry.
NOTE: This answer chose the adjective 'tellurian' for Earthlike as in Earthlike planets.