I don't know who you are talking to at the bank, but they don't know exactly/precisely what they are talking about.
Yes, it has been titled, but in Texas it is an E-Title, this is where the bank employee is confused. There is always an E-Title issued whenever there is a lien. Texas will only print a paper title when there isn't a lien on it, or AFTER the lien has been released by the lender. When a vehicle is sold new, there is an MSO (a kind of paper title issued by the manufacturer) that must be surrendered to the state when it is sold. An E-Title or paper title will only be issued after the MSO is surrendered. The E-Title is in the current owner's name showing the bank as lienholder.
Yes, the bank can and will put it in your name, and if (as in your case) there is not going to be a lien on the title, the state will print a paper title and mail it to you. Yes, both you and the seller (as well as the lender) will need to sign paperwork at the bank (including a 130-U title transfer form) in order for this to happen.
Yes, you WILL need to pay the 6.25% sales tax. Sales tax is collected any time a vehicle (or anything else with a title) changes hands.