The cut function has labels parameter that allows you to make your own labels. There may be a more elegant way to do this but here is my method in this case.
Fechas <- seq.Date(from = as.Date("2020-03-13"), to = as.Date("2020-03-24"), by = 1)
Fechas
#> [1] "2020-03-13" "2020-03-14" "2020-03-15" "2020-03-16" "2020-03-17"
#> [6] "2020-03-18" "2020-03-19" "2020-03-20" "2020-03-21" "2020-03-22"
#> [11] "2020-03-23" "2020-03-24"
Labels1 <- seq.Date(from = as.Date("2020-03-13"), to = as.Date("2020-03-22"), by = 3)
Labels1 <- format(Labels1, "%d-%m-%Y")
Labels2 <- seq.Date(from = as.Date("2020-03-15"), to = as.Date("2020-03-24"), by = 3)
Labels2 <- format(Labels2, "%d-%m-%Y")
LabelsAll <- paste(Labels1, Labels2, sep = " - ")
LabelsAll
#> [1] "13-03-2020 - 15-03-2020" "16-03-2020 - 18-03-2020"
#> [3] "19-03-2020 - 21-03-2020" "22-03-2020 - 24-03-2020"
FechasFactor <- cut(Fechas, breaks = "3 days", labels = LabelsAll)
FechasFactor[1]
#> [1] 13-03-2020 - 15-03-2020
#> 4 Levels: 13-03-2020 - 15-03-2020 ... 22-03-2020 - 24-03-2020
FechasFactor[4]
#> [1] 16-03-2020 - 18-03-2020
#> 4 Levels: 13-03-2020 - 15-03-2020 ... 22-03-2020 - 24-03-2020
Created on 2020-06-29 by the reprex package (v0.3.0)