Since OP has posted the (incorrect) output generated by their code I’m going to assume the awk scripts they’ve posted are not their real code otherwise the escaped brackets would be generating syntax errors, g:

$ awk -F\t 'NR==FNR {a[$2];next} {print $0 FS (($6) in a?"Fail":"Pass")}' test_proteome.tsv test_peptide.tsv
awk: cmd. line:1: NR==FNR {a[$2];next} {print $0 FS (($6) in a?"Fail":"Pass")}
awk: cmd. line:1:           ^ backslash not last character on line
awk: cmd. line:1: NR==FNR {a[$2];next} {print $0 FS (($6) in a?"Fail":"Pass")}
awk: cmd. line:1:           ^ syntax error

Also, to designate a tab as the field delimiter we need to quote the t.

Until OP updates the question with the actual code I’m going to assume we’re starting with the following:

awk -F't' 'NR==FNR {a[$2];next} {print $0 FS (($6) in a?"Fail":"Pass")}' test_proteome.tsv test_peptide.tsv

awk 'NR==FNR {proteome[$2]; next} {status = ($6 in proteome ? "Fail" : "Pass"); print $0 "t" status}' test_proteome.tsv test_peptide.tsv

The second script will fail to generate the desired results because awk will use the default field separator of white space (ie, all spaces and tabs will be processed as field separators). This means the values in the column 2 of test_proteome.tsv are AA_seq, Immunoglobulin and Immunoglobulin.

If we update the second script to awk -F't' 'NR==FNR ...' we find that both scripts now generate:

FusionID    Peptide HLA_Allele  EL-score    EL_Rank AA_seq_full Pass
TUFT1--PBXIP1|chr1:151540426:+|chr1:154951535:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNGTRNWCTLVDVHPEDQAAGRSSRLKALSLAAF*QRRLRSRAPWKVMFVVWSLLAQETQ*SRETCRRPP  Pass
TUFT1--PBXIP1|chr1:151540426:+|chr1:154948366:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNOVEL*PEPTIDE  Pass

A couple issues:

• header line has the string Pass appended on the end (OP hasn’t coded anything to handle the header so the ternary operation applies here, too)
• the first data line ends with Pass instead of Fail

Looking at the data we see there are no matches in the given input data; of particular interest are the strings that start with MNGTRN:

test_proteome.tsv : MNGTRN ... RRPPT
test_peptide.tsv  : MNGTRN ... RRPP

Notice one has a trailing T while the other is missing the trailing T. Since OP has stated we are looking for an exact match, and we have no exact matches, this associated line is appended with Pass.

Since OP’s expected output does not display the trailing T I’m going to assume this was a typo. I’ve removed the trailing T from my copy of test_proteome.tsv.

At this point both of OP’s scripts generate the desired output … except for that Pass in the header line:

FusionID    Peptide HLA_Allele  EL-score    EL_Rank AA_seq_full Pass
TUFT1--PBXIP1|chr1:151540426:+|chr1:154951535:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNGTRNWCTLVDVHPEDQAAGRSSRLKALSLAAF*QRRLRSRAPWKVMFVVWSLLAQETQ*SRETCRRPP  Fail
TUFT1--PBXIP1|chr1:151540426:+|chr1:154948366:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNOVEL*PEPTIDE  Pass

Modifying OP’s 1st script to handle the header line:

########
# do not add to the header line:

$ awk -F't' 'NR==FNR {a[$2];next} FNR==1 {print $0;next} {print $0 FS (($6) in a?"Fail":"Pass")}' test_proteome.tsv test_peptide.tsv
FusionID    Peptide HLA_Allele  EL-score    EL_Rank AA_seq_full
TUFT1--PBXIP1|chr1:151540426:+|chr1:154951535:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNGTRNWCTLVDVHPEDQAAGRSSRLKALSLAAF*QRRLRSRAPWKVMFVVWSLLAQETQ*SRETCRRPP  Fail
TUFT1--PBXIP1|chr1:151540426:+|chr1:154948366:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNOVEL*PEPTIDE  Pass


########
# add 'Result' on end of header line:

$ awk -F't' 'NR==FNR {a[$2];next} FNR==1 {print $0 FS "Result";next} {print $0 FS (($6) in a?"Fail":"Pass")}' test_proteome.tsv test_peptide.tsv
FusionID    Peptide HLA_Allele  EL-score    EL_Rank AA_seq_full Result
TUFT1--PBXIP1|chr1:151540426:+|chr1:154951535:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNGTRNWCTLVDVHPEDQAAGRSSRLKALSLAAF*QRRLRSRAPWKVMFVVWSLLAQETQ*SRETCRRPPT Fail
TUFT1--PBXIP1|chr1:151540426:+|chr1:154948366:-|FRAMESHIFT  GSDGCWGLY   HLA-A01:01  0.79990000000000006 0.1002  MNOVEL*PEPTIDE  Pass

NOTE: I’m assuming the next-to-last field of the last line in OP’s expected output is a typo and the asterisk is not supposed to be escaped, ie, MNOVEL*PEPTIDE should be MNOVEL*PEPTIDE

An example using a bash script in a didactic way, not compressed in a one liner awk command

#!/bin/bash

# Input files
file1="test_peptide.tsv"
file2="test_proteome.tsv"

# Output file
output_file="output.tsv"

# Column numbers to compare
column1=6
column2=2

#write the headers to the output file
printf '%stResultn' "$(head -n 1 $file1)" > $output_file

#for row 2 to the number of lines in the file1
for i in $(seq 2 "$(cat $file1 | wc -l)")
do
    if [ "$(awk -F 't' -v i="$i" -v col1="$column1" 'NR==i {print $col1}' $file1)" 
    == "$(awk -F 't' -v i="$i" -v col2="$column2" 'NR==i {print $col2}' $file2)" ]
    then
        printf '%stFailn' "$(awk -v i="$i" 'NR==i {print $0}' $file1)" >> $output_file
    else
        printf '%stPassn' "$(awk -v i="$i" 'NR==i {print $0}' $file1)" >> $output_file
    fi
done