As long as the interior of the ggplot() call is empty, there shouldn't be a name conflict if you use different data frames with similarly named variables. The conflict comes when defining the infrastructure of the plot with one data frame and then using another data frame with the same variable names in a geom appended to the originally defined aesthetics. *That* causes trouble, and I experienced that while playing with your problem.  For example,
g <- ggplot(dat.scor, aes(x = RDA1, y = RDA2, ....))
g + ... + geom_segment(data = arrows, aes(x = RDA1, y = RDA2...))     # this goes kaboom
However, if you define   g <- ggplot()   and then define the data to be used inside a geom, then that ought to work even if two different data frames have the same variable names; it's kind of like with() in base R.
This is what I did, which is very similar to yours:
# Subset into two data sets, get rid of the redundant Type:
dat.scorspec <- subset(dat.scor, Type == 'Species')[, -3]
dat.scorbipl <- subset(dat.scor, Type == 'Biplot')[, -3]
p <- ggplot()
p + geom_vline(x=0,colour="grey50") +
    geom_hline(y=0,colour="grey50") +
    geom_text(data = dat.scorspec, aes(x = RDA1, y = RDA2, 
              label=rownames(dat.scorspec)), angle=45, size=3,
              colour = 'blue') +
    geom_segment(data = dat.scorbipl, aes(x = 0, y = 0, 
                 xend = RDA1, yend = RDA2), size = 0.5, colour = 'red') +
    geom_text(data = dat.scorbipl, aes(x = RDA1, y = RDA2,
              label = rownames(dat.scorbipl)), size = 5, angle = 45, 
              vjust = 1, colour = 'violet') +
    theme_bw()
I used two geom_text() calls, one for the point labels and one for the biplot axis labels. Notice that I used RDA1 and RDA2 as the variable names in all three geoms that processed data, but the data frames are not the same in all three plots.
If separating some overlapping names is meaningful, I'd either systematically change some of the RDA1-RDA2 coordinates slightly (put them in new variables) or consider some jitter, which might be a problem for those points near the origin.
HTH,
Dennis