Can a road bike BB work on track bike?

The axle will probably be too long pushing the chainring too far out from where it should be which is inline with the sprocket. Most track bikes have a rear spacing of 120mm and use a 107mm axle, compared to 130mm for a road bike, and all sorts of axle lengths depending on the chainring manufacturer, but mostly longer than 107mm.

TLDR: Miche crank may not be compatible with Shimano external bearing BB. The Miche BB is compatible of course and will very likely fit your frame (but bearing quality unknown, and probably not as good as Dura Ace).

If you are going to use the crank with the recommended Miche BB you need to know what frame BB shell width and thread size you have. Majority of modern threaded shells are 'British/English' AKA 'BSA' threads. The BB seems to have a varient for Italian threads as well, in case that's what you have. See here for info on the difference. It's very likely the Miche BB is for a 68mm BB shell as that's the road bike standard, but I could not find that bit of info on the Miche site. You can measure your BB shell easily of course.

If you want to use a Shimano Hollowtech II external bearing BB, then you need to know if the crank is compatible with that standard. There are two parameters: the axle diameter and the total width of the BB when installed. Hollowtech II has 24mm axles and 10mm bearing cups for total width of 88mm (if I recall correctly, might be 11mm bearing cups and 90mm width).

The Miche crank and BB may use a different axle diameter and BB width. I did not see "Hollowtech II compatibility" when looking at the Miche page. You'll need to dig a bit deeper into the specs to find out.

As an aside, I don't know if there is such a thing as a dedicated track bike BB. Most track bikes use a cartridge type bottom bracket with square taper cranks (or one of the splined types I guess). The reason for this I believe is that cartridge BBs offer very good bearing alignment and low friction. This is not guaranteed with external thread in bearings, and press in BB bearings can definitely have very bad problems in this regard. You might want to think about that BTW when considering the Miche crank.

The product page you linked to is listing the crank in question as being compatible with Miche Evo Max bottom brackets, which in different types for different frame interfaces. It will probably also be fine with a Shimano road BB, most 24mm road and track cranks are, but that's not being called out as compatible here, so you're rolling the dice a little.

One of the potential gotchas in a case like this is that this crank is the sort that uses one high-torque crank bolt on the left crank, which means unlike Shimano it does preload via either an elastomer or a wavy washer. If it's a wavy washer and if it doesn't come with the crank, which they sometimes don't, then getting this plus a Shimano BB won't get you everything you need to install it.

Although the question is almost 1 year old now I have a simple answer: no, it doesn't fit.

How do I know? I have tried the exact same (Miche Pistard 2.0 crankset and the same Dura Ace BB). The axle of the crankset it too thick. I managed to get it in, but it was very very tight. When I tried to attach the left arm crank and started the tightening, I noticed something wasn't right. When I tried to remove the right side crank, my BB broke due to the force applied. Lucky everything else remained perfectly fine...

Chainline and everything else will be fine, those are standard.

I don't know what else to get yet. Maybe just stick with the recommended Miche Evo BB