Solid and stranded conductors of the same gauge do have the same current-carrying capabilty. If you look at Table 8, you will see that stranded conductors are slightly larger to make up for the slightly smaller cross-sectional area of stranded, however solid does have slightly less resistance over distance than stranded, but hardly enough to make a big difference on calculations.
They are rated the same. None of the tables in 310 differentiate between solid and stranded. The only cable I know where it differs is with flexible cable designs AKA super-flex.