Can there be more than one airway with the same name in the world?
No.

From ICAO SARPs Annex 11 'Air Traffic Services' Appendix 1:


  3.1.3 A basic designator assigned to one route shall not
  be assigned to any other route.


And to clarify what is meant by designator for the ATS routes (airways): a designator is a letter followed by 1 to 3 digits (1 to 999). In your case the L602 is the designator. The preceding U (for Upper) is a supplementary letter.

RE Is UL602 still part of the L602 airway or are they two separate airways?

They are the same route (lateral path), but not necessarily have the same navaids/waypoints:


The lower ATS route may require an extra VOR, for example, to compensate for the low altitude line-of-sight limitation.
It may also have extra waypoints for joining the SIDs and STARs to the airway (on/off ramp highway analogy). This is more applicable to the example given since it's an RNAV-only ATS route (see letter L below).




Bonus: The current letters for the designators are:


A, B, G, R (regional non-RNAV)
L, M, N, P (regional RNAV)
H, J, V, W (non-regional non-RNAV)
Q, T, Y, Z (non-regional RNAV)


A regional ATS route is usually one that passes through more than 1 FIR.