Is it true that each solution is corresponding to a specific possible universe?
This is looking it backwards. One looks at all these vacua for solutions that fit our current cosmological observations, which is a single one out of this huge number.
If yes, can we identify and pick a single solution that is corresponding to our universe?
Researchers are trying but have not been able to do so. Look at this talk , page 23:
To summarize the status of the string landscape as a “top-down”
concept which leads to candidate vacua, EFT,(effective field theory) and predictions,
Classification and calculational techniques are moderately well
developed but by no means a mature subject.
In addition people have suggested other relevant classes of 4d
vacua (e.g.nonsupersymmetric), and in studying cosmology one
may need to understand vacua with other numbers of large
dimensions. So this is definitely a work in progress.
On the other hand, the string landscape is a mathematically
well-defined structure, someday even rigorously so, and
eventually we will understand it
The link is informative, read from the beginning.