I think they were originally separate unjoined streets. But a road was built up the hill, joining them together. I guess people couldn't agree on who should have their houses renumbered.
If manually specifying the address you probably have enough wiggle room to specify which 10 Something St you mean. I've encountered systems online that will try to normalize and correct street addresses to an official format... I could imagine that these would cause problems.
(And don't confuse those addresses with the 10 New Something St, or 10 Something Rd, both of which are only a few hundred meters away.)