The BTC deposit address I got from GDAX started with a 3. I plugged it into withdraw LTC on Binance. The same exact address exists both as a BTC and LTC address. Also, given that the BTC address is owned by GDAX, can we assume that the LTC address is as well?
At a technical level, this is something that GDAX has the necessary tools and information to recover for you.
However, as a business, this is something that GDAX has intentionally made it very difficult to accomplish for you.
In order to recover these funds, someone at GDAX will need access to the private key associated with the BTC address. They will then need to take the LTC funds that are accessible with that key and send them somewhere for you.
Think about what that means for a moment.
In order to guarantee security of your funds, GDAX has intentionally made it very difficult for any individual in their company to access any of the private keys. That way, individuals in the company don't need to be trusted not to steal their customer's funds. You are asking them to break their own security model. Whomever accesses the private key for you would be able to access other private keys as well, and theoretically could steal funds. As such, they will almost certainly need a complicated process with redundant checks-and-balances involving multiple levels of authorization. The typical GDAX support person isn't going to be able to do this for you. He's going to need to recognize what the situation is and escalate it to the appropriate level within the company to address the problem for you. I'd expect this to take a while, and to incur some costs (they're not going to want to set a precedent of doing all this work every time some user sends $0.10 of crypto to the wrong address). Note that every time they do this, they are effectively creating a temporary hole in their own security. If it was fast and easy, then malicious users would frequently make such "mistakes" in the hopes of finding a way to take advantage of the security hole.