I'm active on another beta site that has migrated several questions to this site, and our usual course of action in cases like this that appear to be blatant advertising attempts is simply to flag the post as spam. You could also flag the post for moderator attention and explain the situation, but flagging for spam allows other non-mods to take a look too, and action may happen quicker this way.
If the user only includes the link on one, fairly well-formulated answer, I'll usually flag it for moderator attention, explain the situation, and write a short comment explaining to the user that they need to disclose any affiliation. However, in cases where the user has just joined and immediately added the link to several old posts, I think it's a safe assumption that they are just trying to advertise. SE's policies may differ from other sites, but I wouldn't think immediately posting your link on multiple old posts is tolerated many places either.
If the link is to a site I recognize, e.g. github, and affiliation is properly disclosed, I usually won't flag it. I assume other users that are more familiar with what's out there will take a look at it too, even if it passes my quick glance.
Of course, my experience is on a different beta site, so we may be more or less tolerant of links than active users on this site. One thing you shouldn't do, however, is edit the post to remove the link. That makes it harder for mods and other users to quickly detect possible spam.
In the specific case you're referring to, it seems pretty clear that the answers are spam because
A cursory google search on the name of the user you're referring to yields a personal website he designed, which links to btc-trade-center.com as a recent item in the news feed. This indicates to me that it's a relatively new site that they're hoping to promote.
The whois record indicates that the domain was just registered two weeks ago. This is further evidence that it's a new site that someone is trying to promote.
A user with the name of the personal website referenced earlier just made an account and posted a similar answer on yet another old question. Even the picture for both accounts is the same, so this clearly looks like someone seeking to advertise their new site.
I would like to hear from high-rep users and/or mods about this as well, just to make sure that I'm articulating the site's policy correctly.