Actually, if you look at the simplified block diagram for the LT3080 (page 1 of the data sheet) it is practically an op amp plus an emitter follower with 100% negative feedback that sets the gain to one.
So in this case, yes, the offset voltage is specifying just the same thing as it would with an op amp. So, if you supply a 1V input to the SET pin then you'll get out a fixed voltage that is between 0.994 V and 1.006 V. It will vary from part to part, but will stay the same for the same part.
The offset will change a little bit with temperature (+/- 500 uV, graphs on page 5 of the data sheet), and with load current (0 to -750 uV), and a negligible amount with output voltage (less than 100 uV over the full range) and will drift a little with time (not specified on the data sheet but probably in the order of 1 to 10 uV per month with a diminishing drift with time, it eventually settles down).