DaysToMatch <- as.Date(c("1991-03-30", "1993-12-18"))

countDays <- function(initialdate, enddate, DaysToMatch) {
   measurementwindow <- seq(as.Date(initialdate), as.Date(enddate), by="days")
   commonDays <- length(intersect(measurementwindow, DaysToMatch))
   #Or sum(DaysToMatch %in% measurementwindow)
   return(commonDays)
}

df$commonDays <- mapply(countDays, df$initialdate, df$enddate,
                 MoreArgs = list(DaysToMatch = DaysToMatch))

df
#  individual initialdate    enddate  RGR commonDays
#1          1  1990-03-01 1991-08-05 0.25          1
#2          2  1991-05-07 1993-02-17 0.30          0
#3          3  1992-05-02 1995-06-22 0.09          1
#4          4  1994-03-02 1995-09-12 0.48          0
#5          5  1995-03-08 1997-08-18 0.34          0