This is the ugliest solution ever. I hope that output is what OP had in mind.
If anyone knows how to improve this code please help.

    pathInput <- "FILE"
    library(data.table)
    library(tidyverse)

   # Read in 
    d <- pathInput %>%
        read_delim(delim = "[[") %>%
        setDT()
   # Have to do this on different object
   # Dont't know why
    d2 <- as.data.table(d[, X1]) %>%
        .[!is.na(V1)] %>%
        .[, col1 := sapply(strsplit(V1, " "), "[[", 1)] %>%
        .[, col2 := sapply(strsplit(V1, " "), "[[", 2)] %>%
        .[, col3 := sapply(strsplit(V1, " "), "[[", 3)]
   # Where new table starts
    foo <- grep("NAME", d2$V1)
   # Write to list
    res <- list()
    for(i in 1:length(foo)) {
        if (i < length(foo)) {
            res[[i]] <- d2[(foo[i] + 1):(foo[i + 1] - 1), .(col1, col2, col3)]
        } else {
            res[[i]] <- d2[(foo[i] + 1):nrow(d2), .(col1, col2, col3)]
        }
    }

res returns:

[[1]]
          col1        col2      col3
1: FBgn0037415 FBgn0037415 0.8035441
2: FBgn0010812 FBgn0010812 0.6579683
3: FBgn0265351 FBgn0265351 0.6443309

[[2]]
          col1        col2      col3
1: FBgn0037227 FBgn0037227 0.9997242
2: FBgn0040682 FBgn0040682 0.9997242

[[3]]
          col1        col2      col3
1: FBgn0026620 FBgn0026620 0.5241095
2: FBgn0263619 FBgn0263619 0.5420427
3: FBgn0263353 FBgn0263353 0.9812295
4: FBgn0037424 FBgn0037424 0.9793901
5: FBgn0037428 FBgn0037428 0.9779420
6: FBgn0037430 FBgn0037430 0.9540148
7: FBgn0004777 FBgn0004777 0.8962534
8: FBgn0004778 FBgn0004778 0.9810570

From here you can create your tidy tables.

Another cleaning version:

# read as lines, every line is a character
x <- readLines("myFile.txt")

# split on "[["
x <- split(x, cumsum(grepl("[[", x, fixed = TRUE)))

# tidy up
clean <-
  lapply(x, function(i){
    # get list id
    listID <- as.numeric(gsub("\\D+", "", i[1]))
    # column names
    header <- unlist(strsplit(gsub("\\s+", " ", trimws(i[2])), " "))
    # split on " " and convert to dataframe
    res <- as.data.frame(do.call(rbind, strsplit(tail(i, -2), " ")))[, 2:3]
    # add name and list id
    colnames(res) <- header
    res$ListID <- listID
    res
  })

pletny of way to do that I believe depending how you want to do it. Take a look at the similar solutions in stackoverflow.

Link1

Link2

Link3

You can come up with a solution from there. However this is not a bioinformatics question tbh. So try to use stackoverflow for such queries.

Can you not solve that using the split command? Something like this:

    # Read the data:
        a <- list(read.table(sep="\t", text = "rows NAME    MEM.SHIP
        FBgn0026620 FBgn0026620 0.5241095
        FBgn0263619 FBgn0263619 0.5420427
        FBgn0263353 FBgn0263353 0.9812295
        FBgn0037424 FBgn0037424 0.9793901
        FBgn0037428 FBgn0037428 0.9779420
        FBgn0037430 FBgn0037430 0.9540148", header=T), 
        read.table(text="rows   NAME    MEM.SHIP
        FBgn0037415 FBgn0037415 0.8035441
        FBgn0037430 FBgn0037430 0.6579683
        FBgn0265351 FBgn0265351 0.6443309", sep="\t", header=T))

    # Combine the lists:
    b <- do.call(rbind, a)

    # Extract MEM.SHIP info

    sapply(split(b, b$NAME), function(x) x["MEM.SHIP"])

    # Which give you this:

        $FBgn0026620.MEM.SHIP
        [1] 0.5241095

        $FBgn0037424.MEM.SHIP
        [1] 0.9793901

        $FBgn0037428.MEM.SHIP
        [1] 0.977942

        $FBgn0037430.MEM.SHIP
        [1] 0.9540148 0.6579683