The short answer is No, there is no way to identify you just by a simple connection to a WiFi. But, there are some things you may have in mind in order to complete this short answer... Let´s see:
1 - As you said the IP address does not identifyany device in a unique way. There is another address that do that, this is the MAC address. This one is in each networking device (as the WiFi device your smartphone may have onboard) and it is unique between each device because each vendor has a code and they are supposed to assign unique values to each device they produce. BUT, in practice this don´t happen, becuase some MACs are repeated (this is very strange to find but there is some very little probability) and you can clone the MAC of another device with some software tools, so the MAC address is not a practical unique identifier of network devices. Moreover, there is no relation between the MAC and you. I mean, when you buy a device there is no association between the MAC and the client who has bought the device.
2 - There are a lot of hacking techniques that may be applied to identify you in a network. They are no easy to do and need to be carefully executed by someone that has the knowledge and the time to do it. I mean, it´s not as in the films that the hacker takes the keyboard and press 100 keys per second and voila! he can hack the FBI. This techniques need a lot of work to successfully work.
3 - Using encrypted traffic will always turn very difficult to identify you. For example, when you are browsing the web it´s important that you can see a little green padlock and the HTTPS protocol (for example, https://www.google.com and not the insecure way: http://www.google.com). Because, if someone is sniffing the connection may find some important data on your transmissions. Nowadays, almost every website us HTTPS so you don´t have to worry. Also, WhatsApp encrypts the traffic. This is important, because, let´s say you are writing an email and when you send it the connection is not encrypted, so a "hacker" that is sniffing your traffic would be able to see your sign in the email´s footer and identify you.
In conclusion, there is no direct way to identify you in a network if you don´t name your device with your real name. BUT, it is also true, that someone with the correct knowledge and with some techniques successfully applied may know who you are.
Hope it helps!
1They could rule out all other devices one by one, or correlate your arrival with a device appearing on the network. Devices have MAC addresses which will often give away the manufacturer of your WiFi chip and would also uniquely identify your device. If they needed to then they could take everyones phones and check whether the MAC address matches. Your phone may also broadcast an identifier, not necessarily your name, that could be used to uniquely identify your device. Word to the wise: don't use your phone to browse dubious websites on work WiFi. – Mokubai – 2017-07-30T15:37:36.340
Why downvote? Seems like a perfectly reasonable question to me.. – DividedByZero – 2017-07-30T17:00:33.867
You may hide your MAC address by spoofing it. Change the browser useragent to hide your phone model. – Biswapriyo – 2017-07-30T17:22:24.987