I think this is xy problem. With example data above and expected output, try the following code:

> df
     Column1    Column2      Column3
1 gene=LOC11  product=A partial=true
2 gene=LOC11  product=A partial=true
3 gene=LOC22  product=B partial=true
4 gene=LOC33  product=C partial=true
5  gbkey=CDS gene=LOC44    product=D
6  gbkey=CDS gene=LOC55    product=E
7  gbkey=CDS gene=LOC66    product=F
> library(dplyr)

If you want old columns to be edited:

> df %>%
+   select(1,2) %>% 
+   mutate(across(.cols = everything(), ~ifelse(grepl("gene",.), .,"NA")))
     Column1    Column2
1 gene=LOC11         NA
2 gene=LOC11         NA
3 gene=LOC22         NA
4 gene=LOC33         NA
5         NA gene=LOC44
6         NA gene=LOC55
7         NA gene=LOC66

if you want new columns to be created:

> df %>%
+   select(1,2) %>% 
+   mutate(across(.cols = everything(), 
+                 ~ifelse(grepl("gene",.), .,"NA"), 
+                 .names = "new_{col}" ))
     Column1    Column2 new_Column1 new_Column2
1 gene=LOC11  product=A  gene=LOC11          NA
2 gene=LOC11  product=A  gene=LOC11          NA
3 gene=LOC22  product=B  gene=LOC22          NA
4 gene=LOC33  product=C  gene=LOC33          NA
5  gbkey=CDS gene=LOC44          NA  gene=LOC44
6  gbkey=CDS gene=LOC55          NA  gene=LOC55
7  gbkey=CDS gene=LOC66          NA  gene=LOC66

In your starting file, does gene=LOCXX always appear in either column 1 or column 2? How I would approach it is to split your dataframe into two (or more) depending on which column gene=LOCXX appears. You can then limit each dataframe to the two columns you want then rbind them back into a single dataframe.

df <- tibble(col1 = c("gene=LOC11","gene=LOC22","gene=LOC33","gbkey","gbkey","gbkey"),
             col2 = c("A","B","C","gene=LOC44","gene=LOC55","gene=LOC66"),
             col3 = c("partial","partial","partial","D","E","F"))

df1 <- df[grep("LOC",df$col1),] %>% select(1,2)
df2 <- df[grep("LOC",df$col2),] %>% select(2,3)
colnames(df2) <- colnames(df1)
rbind(df1, df2)

A tibble:6 x 2
col1
<chr>
col2
<chr>
gene=LOC11  A           
gene=LOC22  B           
gene=LOC33  C           
gene=LOC44  D           
gene=LOC55  E           
gene=LOC66  F

Just for your knowledge, the reason why your code is failing is because of this line:

mutate(df, columnA = grep("LOC.*", df$Column1, value = TRUE)

You are asking to add a column to your dataframe that is smaller than your dataframe and mutate doesn't know how to handle that. I think you're trying to make a column with boolean values. If that's the case, I'd look into the case_when function:

df %>% mutate(ColA = case_when(str_detect(col1, "LOC") == TRUE ~ 1,
                               str_detect(col1, "LOC") == FALSE ~ 0))

This can be modified to add the "gene=LOCXX" as that column like this:

df %>% mutate(ColA = case_when(str_detect(col1, "LOC") == TRUE ~ col1,
                               str_detect(col2, "LOC") == TRUE ~ col2))

col1
<chr>
col2
<chr>
col3
<chr>
ColA
<chr>
gene=LOC11  A   partial gene=LOC11  
gene=LOC22  B   partial gene=LOC22  
gene=LOC33  C   partial gene=LOC33  
gbkey   gene=LOC44  D   gene=LOC44  
gbkey   gene=LOC55  E   gene=LOC55  
gbkey   gene=LOC66  F   gene=LOC66