Very few motorbikes run on 6 V any more. 12 V makes a lot of sense if there is a started motor.
Certainly the current rating of the battery isn't a problem. There are small loads like panel light bulbs that are rated at 12 V any only 0.1 A connected to the same battery that runs the starter motor.
A fuse is a good idea, and will stop wires burning out if the regulator shorts out, but otherwise the current taken will only be a little more than what the microcontroller takes.
There are better regulators than the 7805. The quiescent current, the current that it takes all the time even with no load, is fairly high and so you shouldn't use it if it will be running when the bike it turned off.
The drop-out voltage is about 2.5 V, so you won't get 5 V out if the voltage is less than 7.5 V, so it could drop when starting.
The maximum input voltage for a 7805 isn't large enough to survive the spikes. It will certainly die if there is a significant load dump. In my experience, motorbikes are worse than cars for noise on the power lines.
There are regulators designed for automotive use, which might well work better.