Cell phones sometimes expose their data connection as an old AT-style modem. I'd say on older cell phones this is going to work closer to how you'd expect, i.e. you put the phone into modem mode, you either connect A) an actual serial cable (or a proprietary cable that exposes the UART pins on the phone), B) a USB cable that's functioning as a COM port, or C) a Bluetooth virtual COM port, and use it just as you would a normal external serial modem.
So the AT conversation you are having, which is not working, is the AT command interface bolted on to the cell phones data connection. Presence of an AT command interface does not automatically mean you are talking to a device that does analog-style modulation/demodulation. Many phones used to do this. With most Android phones now supporting RNDIS/USB over Ethernet it's not very necessary anymore and I'm not sure if many do or the current state of this.
As an aside, I think there are extensions to the AT command set to specifically support cell phones sharing data via this interface, and even to support things like SMS.
No cellphone does the analog-style modulation/demodulation that an analog modem does natively that I've ever heard of. You might be able to get softmodem
or similar working on an Android device if you cross-compile it for ARM, but I don't know if there is any application like that for Blackberry; I highly doubt one exists.