R fuzzy string match to return specific column based on matched string

You are 90% of the way there…

You say you want to

know with which row of data the string was matched from df2

You just need to understand the code you already have. See ?amatch:

amatch returns the position of the closest match of x in table. When multiple matches with the same smallest distance metric exist, the first one is returned.

In other words, amatch gives you the index for the row in df2 (which is your table) that is the closest match of each address in df1 (which is your x). You are prematurely wrapping this index by returning the new address instead.

Instead, retrieve either the index itself for lookup or the unique_id (if you are confident that it is truly a unique id) for a left join.

Illustration of both approaches:

library(data.table) # you forgot this in your example
library(stringdist)
df1 <- data.table(Address1 = c("786, GALI NO 5, XYZ","rambo, 45, strret 4, atlast, pqr","23/4, 23RD FLOOR, STREET 2, ABC-E, PQR","45-B, GALI NO5, XYZ","HECTIC, 99 STREET, PQR","786, GALI NO 5, XYZ","rambo, 45, strret 4, atlast, pqr"),
                  Year1 = 2001:2007) # already a vector, no need to combine
df2 <- data.table(Address2=c("abc, pqr, xyz","786, GALI NO 4 XYZ","45B, GALI NO 5, XYZ","del, 546, strret2, towards east, pqr","23/4, STREET 2, PQR","abc, pqr, xyz","786, GALI NO 4 XYZ","45B, GALI NO 5, XYZ","del, 546, strret2, towards east, pqr","23/4, STREET 2, PQR"),
                  Year2=2001:2010)
df2[,unique_id := sprintf("%06d", .I)] # use .I, it's neater

# Return position from strVec of closest match to str
match_pos = function(str, strVec, n){
  amatch(str, strVec, method = "dl", maxDist=n,useBytes = T) # are you sure you want useBytes = TRUE?
}

# Option 1: use unique_id as a key for left join
df1[!is.na(Address1) | nchar(Address1>0), # I would exclude only on NA_character_ but also empty string, perhaps string of length < 3
    unique_id := df2$unique_id[match_pos(Address1, df2$Address2,3)] ]
merge(df1, df2, by='unique_id', all.x=TRUE) # see ?merge for more options

# Option 2: use the row index
df1[!is.na(Address1) | nchar(Address1>0),
    df2_pos := match_pos(Address1, df2$Address2,3) ] 
df1[!is.na(df2_pos), (c('Address2','Year2','UniqueID')):=df2[df2_pos,.(Address2,Year2,unique_id)] ][]