How to catch and name Tibbles using a for-loop from a larger data frame.

A loop is not the best way to accomplish this task, take a look at this alternative method

library(dplyr)

x <- read.csv("BOM_test.csv")

x %>% 
    nest_by(Assembled_Product_Code) %>% 
    mutate(data = set_names(list(data), Assembled_Product_Code)) %>%
    list2env(x = .$data, envir = globalenv())

Note: Next time please provide a proper REPRoducible EXample (reprex) illustrating your issue.