If you want to extract individual components, try str_match:
test <- c("aaa[name1,1]", "bbb[name2,3]", "ccc[name3,3]")
stringr::str_match(test, "([a-zA-Z0-9_]+)[[]([a-zA-Z0-9_]+),.*?(\\d+)\\]")
## [,1] [,2] [,3] [,4]
## [1,] "aaa[name1,1]" "aaa" "name1" "1"
## [2,] "bbb[name2,3]" "bbb" "name2" "3"
## [3,] "ccc[name3,3]" "ccc" "name3" "3"
Otherwise use str_extract.
, [ ERE/TRE , [, .. [[].
, , str_match_all str_extract_all.