This is not an answer to your question: Is there any policy in effect to ensure the uniqueness of user names and if not, why?1
But it seems that you are also interested in this question: "I saw a user on MO. Can I find his math.SE account?" (So perhaps you will forgive me that I am answering something different from what you actually asked, since this seems of interest to you, too.)
It seems that you are talking about this question: Nonlinear Differential Equation System Example (It is possible that the question is eventually going to be deleted - at least if it stays closed and with negative score, the question will be removed by roomba in a month. But at the moment everybody can access it; if it is deleted, users with 10k+reputation can still display the question after clicking on the link.)
If you go to the question and click on the OPs name, you can see their MathOverflow profile: https://mathoverflow.net/users/117772/daniel On the left side you can see what other accounts he has, in this case there is only StackOverflow account displayed. If the same user creates account on this site, the new account will be displayed there, too. You can also view accounts on the active tab. And by clicking on the word accounts you get to the network profile, where all accounts are displayed. (This is useful if there are many accounts.)2
You can try with my account - just click on my username and you can try what I described above. You will see a few accounts, but to display all my accounts on various Stack Exchange sites, you have to go to the network profile. (Or you can try the same with your own profile.)
Warning: Users can decide not to display some of their accounts there. Details are explained here: How do I hide my accounts on a specific Stack Exchange site. They may have various reasons for this. But I do not think they are many users who do this. And I find this somewhat unlikely in the case of relatively new users - knowing that it is possible to hide some accounts requires a bit experience with Stack Exchange system.
So the approach described above will not lead you to Mathematics account of user who has account on this site but decided to hide it from their profile. But in most cases, you should be able to find the user's profile on Mathematics (or on other sites) in this way.3
1As mentioned in the comments, you can already find a few posts on this meta about this issue: Users with the same name. Is this possible?, Nonunique usernames? And, of course, also on meta.SE: Why does Stack Overflow allow duplicate display names? or Are usernames required to be unique?
2I will also add that on the user's MO profile you can see when they were on MathOverflow the last time. (Next to the "last seen" you can see some date or time and if you hover above it you get the exact timestamp.) At the moment where I am posting is, you can see that the user was not online after the closure of the question. (In fact, they did not even see any of the comments there, unless they looked while not logged in.) So if you want to interact with this user or see whether they posted something since the closure of the question on MO, you will still have to wait a bit. I will also mention that on the network profile you can see the user recent actions (comments, posts, ...) on all Stack Exchange sites.
3The situation would be a bit more complicated if you want to contact some user in chat and the user has a very common name. In chat I am not really sure how to think of some other way of finding a specific user (for whom I know account on the main site) other than by searching for the username. However, in the present case this is a non-issue, since the users has reputation below 20.