We can , One exchange just proved it to me . I sent them numerous address and they told me which wallet/exchange the address belongs to .
They do this to not to send BTC to darknet and blacklisted addresses. So i am being curious that how it works.
This doesn't make any sense to me, especially the last part. If some address belongs to an exchange, that is not that hard to find out if you do some data analysis and if you know at least some addresses of that exchange. This info isn't in the address or any particular place on it's own. It is just some research and inside knowledge. As for the wallet software that generated them, I guess there might be wallets that try to generate addresses in a certain recognizable way, but this isn't the case for any popular wallet I know, it is just harder to do so and there is no benefit, in fact it is at the cost of anonymity. It is harder to generate addresses that don't look random and you need to make an effort to do so.