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.
They surely checked Walletexplorer
https://www.walletexplorer.com something that everyone can do, you just need to enter the BTC address. However, you can find only wallets related to exchanges, web wallets, pools, gambling sites.
But you can't identify a wallet software (unless you find it on the www like for example the owner posted in publicly of course)