Create list of all connections between vertices defined by id columns and row number (function f). At the end you are interested only in connection between rows.

f <- function(vec){
  
  i <- last(vec)
  vec <- head(vec, -1)
  
  c(
    seq_len(length(vec) - 1) %>% map(~vec[.x:(.x+1)]),
    vec %>% map(~c(i, .x))
  ) 
}

df$desired_output <- df %>%
  select(matches("^id_[0-9]+$")) %>%
  mutate(row = row_number()) %>%
  pmap(~f(c(...))) %>%
  flatten() %>%
  reduce(rbind) %>%
  igraph::graph_from_edgelist() %>% 
  components() %>%
  membership() %>%
  .[as.character(seq_len(nrow(df)))]

edit

Imagine connections between ids. You are interested in connections between rows. For that you need to add vertices for each row. Those vertices are connected to all ids in that row.

Example for 6th row:

6  EFG   5A   Z5

we are interested in connections between ids (first part in c in function f:

[[1]]
[1] "EFG" "5A" 

[[2]]
[1] "5A" "Z5"

and connections between row and ids (second part of c in f):

[[1]]
[1] "6"   "EFG"

[[2]]
[1] "6"  "5A"

[[3]]
[1] "6"  "Z5"

when you create graph that way you end up with:

and you are interested which row vertices are connected

note

you can use directed = FALSE when creating graph for this result, or mode = "strong" in components if you are interested in that.

Updated Solution In case of only getting the desired column it could be another alternative:

library(dplyr)
library(purrr)
library(igraph)

df %>%
  select(starts_with("id")) %>%
  pmap_dfr(~ as.data.frame(t(combn(c(...), 2)))) %>%
  graph_from_data_frame(directed = TRUE) %>%
  components() %>% 
  groups() -> lst
  
df %>%
  rowwise() %>%
  mutate(grp = seq_len(length(lst))[map_lgl(lst, ~ id_1 %in% .x)])

# A tibble: 15 x 5
# Rowwise: 
   id_1  id_2  id_3  Name              grp
   <chr> <chr> <chr> <chr>           <int>
 1 ABC   1A    Z3    StackOverflow1      1
 2 ABC   2A    Z2    StackOverflow2      1
 3 BCD   3A    Z1    StackOverflow3      2
 4 CDE   1A    Z4    StackOverflow4      1
 5 DEF   4A    Z1    StackOverflow5      2
 6 EFG   5A    Z5    StackOverflow6      3
 7 GHI   6A    Z5    StackOverflow7      3
 8 HIJ   8A    Z6    StackOverflow8      3
 9 IJK   9A    Z7    StackOverflow9      4
10 JKL   10A   Z8    StackOverflow10     5
11 GHI   7A    Z6    StackOverflow11     3
12 KLM   12A   Z8    StackOverflow12     5
13 LMN   13A   Z9    StackOverflow13     6
14 MNO   14A   Z9    StackOverflow14     6
15 NOP   15A   Z1    StackOverflow15     2

In order for you just to plot it:

library(dplyr)
library(purrr)
library(igraph)

df %>%
  mutate(id = row_number()) %>%
  select(starts_with("id"), id) %>%
  pmap_dfr(~ as.data.frame(t(combn(c(...), 2)))) %>%
  graph_from_data_frame(directed = TRUE) %>%
  plot()

You can try the code below

transform(
  df,
  GRP = membership(
    components(
      graph_from_data_frame(
        reshape(
          df,
          direction = "long",
          idvar = c("id_1", "Name"),
          varying = 2:3,
          v.names = "to"
        )[c("id_1", "to")]
      )
    )
  )[id_1]
)

which gives

   id_1 id_2 id_3            Name GRP
1   ABC   1A   Z3  StackOverflow1   1
2   ABC   2A   Z2  StackOverflow2   1
3   BCD   3A   Z1  StackOverflow3   2
4   CDE   1A   Z4  StackOverflow4   1
5   DEF   4A   Z1  StackOverflow5   2
6   EFG   5A   Z5  StackOverflow6   3
7   GHI   6A   Z5  StackOverflow7   3
8   HIJ   8A   Z6  StackOverflow8   3
9   IJK   9A   Z7  StackOverflow9   4
10  JKL  10A   Z8 StackOverflow10   5
11  GHI   7A   Z6 StackOverflow11   3
12  KLM  12A   Z8 StackOverflow12   5
13  LMN  13A   Z9 StackOverflow13   6
14  MNO  14A   Z9 StackOverflow14   6
15  NOP  15A   Z1 StackOverflow15   2

Data

> dput(df)
structure(list(id_1 = c("ABC", "ABC", "BCD", "CDE", "DEF", "EFG", 
"GHI", "HIJ", "IJK", "JKL", "GHI", "KLM", "LMN", "MNO", "NOP"
), id_2 = c("1A", "2A", "3A", "1A", "4A", "5A", "6A", "8A", "9A",
"10A", "7A", "12A", "13A", "14A", "15A"), id_3 = c("Z3", "Z2",
"Z1", "Z4", "Z1", "Z5", "Z5", "Z6", "Z7", "Z8", "Z6", "Z8", "Z9",
"Z9", "Z1"), Name = c("StackOverflow1", "StackOverflow2", "StackOverflow3",
"StackOverflow4", "StackOverflow5", "StackOverflow6", "StackOverflow7",
"StackOverflow8", "StackOverflow9", "StackOverflow10", "StackOverflow11",
"StackOverflow12", "StackOverflow13", "StackOverflow14", "StackOverflow15"
)), row.names = c(NA, -15L), class = "data.frame")