1. awk -F '|' '/^>/ {printf("%s%s,%s\t",(N>0?"\n":""),$3,$0);N++;next;} {printf("%s",$0);} END {printf("\n");}' input.fa : 

Prints 3 fields - string to be compared, sequences header and sequence it self. First delimiter is comma, second one is tab, in the output. While consuming input, field delimiter is "pipe |" and line should start with > (fasta header).

2.  sort -t, -k1,1 -u : 

uses comma as delimiter, sorts on first field and prints unique lines based on first field

3.  cut -d, -f2-   

Cuts field 2 onwards and delimiter for cutting is comma (this would keep sequence header and sequence)

4.  tr "\t" "\n" :  

Replace tab with new line.

It can be further tightened:

$ awk -F '|' '/^>/ {printf("%s%s,%s\t",(N>0?"\n":""),$3,$0);N++;next;} {printf("%s",$0);} END {printf("\n");}' test.fa | sort -k1,1 -u | awk -F'\t' -v OFS="\n" '{print $2,$3}'

However this can be tricky as you would not have control over which sequence to be included (for eg. short vs long). In those cases, I would suggest datamash.

Let us say you would like to extract larger sequence (between/among duplicate records), use this (assuming that fasta sequences are in single line):

$ awk -F '|' -v OFS="\t" '/^>/ {getline seq} {print $3,$0,seq, length(seq)}' test.fa | datamash  -fs -g1 max 4 | awk -F "\t" -v OFS="\\n" '{print $2,$3}'

Let us say you would like to extract smaller sequence (between/among duplicate records), use this:

$ awk -F '|' -v OFS="\t" '/^>/ {getline seq} {print $3,$0,seq, length(seq)}' test.fa | datamash  -fs -g1 min 4 | awk -F "\t" -v OFS="\\n" '{print $2,$3}'

However, above code (from my post) works only if sequences are single line. For converting multi-line fasta records to single line fasta records (flattened format) , there are awk scripts or you can use seqkit.