There is no hard and fast answer about whether or not this user is doing anything wrong. It mostly comes down to why they have this second account, and what they do with it.
This is known as having a sock-puppet, and it is addressed on Meta Stack Exchange by an SE employee. To quote the most important parts:
When should sockpuppets be considered a problem?
There are a handful of legitimate reasons to maintain multiple accounts. A good rule of thumb for identifying abusive socks is: if the second account allows you to do something on the site that your normal account would be prevented from doing, it is abuse. Examples of this include (but are not limited to):
- Voting on your own posts
- Answering your own questions with the other account(s)
- Casting multiple votes on others' posts
- Supporting your own arguments ("+1: shog is right, don't know why the rest of you don't realize this")
- Using bounties to circumvent the rep cap
- Circumventing suspensions, quality bans, or the rate limits on posting questions / answers / comments / etc.
If this user is doing any of these things, this is a serious problem. He should delete his account immediately. If he doesn't, you should flag one of his posts with in need of moderator intervention, and explain the situation with links to both accounts.
Since I'm not a moderator, I can't say for sure what would happen in this scenario, but it's very likely that his second account will be deleted, and his first account temporarily suspended.
Now for what it's worth, there are many users with sock-puppets that don't cause issues, and he might have a sock-puppet for perfectly valid reasons. To quote Shog9 again:
I've probably had a half dozen or so alternate accounts over the course of the site. They're useful sometimes: testing bugs that only show up at low rep levels, reminding yourself what the site "feels like" for someone with only the basic abilities, maintaining a highly unnatural q/a ratio... That said, I always treat these as disposable accounts, and some of them have been removed (either automatically or via mod intervention) when I crossed a line somewhere - you should never assume that a sockpuppet is "protected" if you're actively using multiple accounts on one site; it's entirely too hard to be certain you're not interacting in some inappropriate fashion.
I know that lots of users here have created sock-puppets so that they can create a chatbot, or chat relay service. That's another legitimate use.