Its actually even easier to uncover someones true identity now than it was previously. With all the social media activity a normal person undertakes, you won't even need to access government agencies to be able to find their identity.
Bugging a Device
This is a pretty standard move you see in movies and TV shows all the time. Simply plug in a usb like device onto their computer or laptop, or install an app on their phone which will provide you access to their social media accounts. From there you can link together their social activity, bank details (if saved to their phone), pictures and so on.
Is it possible to purchase information on a person from Social media platforms. News corporations will do this with celebrities who might accidentally post a not so savory photo. Basically anything you put onto a social media platform might be purchasable and often, doesn't belong to you anymore (you hand them the copyright when posting it, although this might have changed recently). Your detective could simply try and purchase additional information on the suspect.
Almost everyone here would have at some point had their credentials stolen. Unless you are a security freak with double authentication on every account, different passwords for every site and you rotate them every couple of months, chances are, at least one of your accounts has been compromised. Its possible for you to purchase these account details online and simply login on as the person you are following. For the less tech savy, they will likely be extremely vulnerable to this, and it should be fairly simple for you to find the corresponding password once you find their username (just have a look when they login).
A good old fashion break in. Most people are working during the day and during this time, you can break into their houses. As much as we would like to believe our neighbors care about our safety, most people would just ignore it. Even more so if you dress as a utility worker. Most locks are pretty easy to pick, and breaking the door or window is also plausible. Once you have access to their home, get your info and get out. Leaving your break in undetected is a bit harder, but not impossible.