When I was young I was taught to conclude my confession with, "For these and all the sins of my past life, I ask pardon and absolution." This is a strange expression and almost seems to imply I've had other lives in the past, does it not?

​No, it does not. This is a mode of speaking, an expression, and should not be understood in a strictly literal manner. There are all sorts of expressions and manners of speaking which, if read in a literalist manner, make little sense, but everyone knows what they mean euphemistically. 
 
For example, a mother may say to her child, “Put on your shoes and socks.” Literally, this would be difficult and clumsy, since it seems to say that I should put on my shoes, and then my socks on top of the shoes. But of course, it means no such thing. And while it is more true to say one ought to put on his socks and shoes, everybody knows what it means. Another example is the expression “coming and going.” But of course one cannot really come, until they first go. So the expression more accurately should be “going and coming.” But despite poor word order, everybody knows what “coming and going” means and adjusts. 
 
And thus, when we ask forgiveness for the sins of our “past life,” it is clear we are referring merely to the sins we have committed in the past. If this saying is bothersome to you, then you may amend it, for it is not a formal or prescribed way of ending the confession, but it is simply a common sentence many use to tell the priest they are done mentioning their sins. 
 
Theologically, one is not required to ask forgiveness for sins of the past which have already been forgiven in the sacrament. And thus, another way a penitent can end his confession is to say something like “For these, and other sins I cannot recall, I ask for pardon and absolution.”