Yes. There is an Organizationally Unique Identifier in your MAC Address that can identify your device (i.e. android, motorola, samsung). Levels of granularity depend on whatever device is actually watching the network traffic (e.g. you might have a firewall that automatically resolves the OUI to a particular vendor).
A lot of it depends on where in the network(s) the "listening" device you're referring to sits. If you jump on the WiFi at a hotel or business, for instance, they will probably have a wireless controller that records your local IP and MAC address. They'll be able to tell from this information what kind of phone you're using, and any traffic that passes through their network to the Internet (via your local IP). Also there are other indications there, such as DHCP traffic. When you first jump on the network, you're going to request an IP address, as a broadcast. That broadcast will reach a DHCP server which will identify you by MAC (and the OUI) and then give you an IP.
Also it's very easy to resolve an IP address to a particular computer. It depends on where in the network you sit. If you're visiting a web site from your phone, it's going to exit your border gateway device using PAT, and the public IP will be used to identify the traffic. If you're on the same network as the "listening" device, it will be able to see your MAC address and OUI, and internal IP, and it can be very easy to pinpoint which originating device that is.
Either way, if it can determine that, how can I change it so that the smartphone can't be.
You'll want to look into MAC address spoofing. Creating a new virtual network device and assigning it a different MAC address is possible on most computers, but on a smartphone (especially an iPhone) it may prove quite difficult.