There's any number of ways that your company's network administrator may have identified you. The principal thing you need to realize is that, while you may be using your own device, you are on your company's network. That grants the company a lot of visibility to the traffic your device generates, should they choose (as it appears they have) to examine it. It also means that you are responsible for complying with your company's policies regarding efficient and appropriate use of their resources (e.g. bandwidth).
The easiest way they could have identified you is by your device's hostname. If it's called "Chuck's iPhone", and you're the only "Chuck" in the office, it's not a huge leap to figure out whose device it is. Changing your hostname is perhaps the simplest thing you can do to obfuscate your identity.
Another way to identify the device is by registration. Your company may have forced you to register via a web portal when you joined the WiFi network. At that time, your iPhone's MAC address could have been logged. Though it's possible to change a device's MAC address, it's rarely done. So it's still a fairly reliable identifier, if you have data to associate it to a person. I'm not aware of any built-in mechanisms to change the iPhone's MAC address, but I've heard there's supposed to be a feature coming in newer iOS versions.
If your phone is making any connections to other corporate systems (e.g. e-mail) while you're on their WiFi, your connections to those systems will be logged. That will associate your company username, which is associated with your identity, to your iPhone's current IP address. The first step to avoid this is to stop using company applications on your phone when you're on the company's WiFi network. (Also make sure that you don't have any apps installed or configured in such a way that they might do so in the background.) The next step, after you've done that, is to get a new IP address. Since you don't know how the DHCP server is configured, the best way to do this would be to disconnect from WiFi, delete the WiFi profile, change your MAC address (if possible), rebuild the WiFi profile, and reconnect. Then, check to make sure that you got assigned a different IP.
The last way they could identify you, which is generally the most difficult to mitigate (especially on a non-jailbroken iPhone), is by direct analysis of your network traffic. Smartphone applications send a lot of data in the clear, both when you're actively using them and when they're running in the background. This data can contain any number of things that might be used to associate the phone to you, personally. It could have your name, user account IDs for online services (which may be based on your name), names of your friends/family, pictures of yourself or friends/family, phone numbers, addresses, or various other flavors of PII. Unless you've personally vetted how every application on your iPhone handles your data, there's no way to know for sure what information might be readily available to the administrators of any network your iPhone connects to. The only way to avoid this is to route all of your iPhone's traffic through a secured VPN gateway. These usually are paid services, and may or may not support your device.
In the end, we come back to what I said in the first paragraph. The company's network is not yours to use however you choose. Especially now that you've been warned, any attempts you make to circumvent their management systems will only further strengthen a disciplinary case against you. Also, just because you might be able to make yourself unidentifiable via network traffic from the data-link layer or above, there are no measures you can take to prevent a network administrator from identifying you at the physical level. With enough access points acting as sensors, network administrators can isolate your device's location to within several meters and follow it as it moves around. This can be used to build an activity profile which could be roughly correlated against your personal work and meetings schedule. To further narrow it down, a tablet or other man-portable device can be used to find your device's location within a few feet.
In short, the best way to keep your personal device from being identified on the company's network is to not use it on the company's network. Keep your personal devices and Internet activities off of corporate hardware, and your network administrators won't care what you do with them.