The key step is renaming the sequence of each file with the gene name in the file name. seqkit is used for renaming.

cat gene1.fasta 
>seq1
aaa

for f in *.fasta; do \
    gene=${f%.fasta}; \
    seqkit replace -p ".+" -r $gene $f; \
done > result.fa

cat result.fa 
>gene1
aaa
>gene2
ccc

For Windows users:

brename -q -l -p "fasta$" | rush -k "seqkit replace -p .+ -r {%:} {}" > result.fa

Where brename is for finding and listing files, rush is for batch operation and {%:} is for extracting the gene name.

what is inside each fasta file? Can you see if this works for you:

with awk:

$ awk '/^>/{sub(".*",">"FILENAME)}1' *.fa | sed -r '/^>/ s/\.fa$//'

with parallel:

$ parallel seqkit replace -p '.+' -r {.} {} ::: *.fa

Replace .fa with appropriate file extension.

Using Linux/Unix (e.g. OSX)

cat gene1.fasta gene2.fasta gene3.fasta > gene.fasta

If you've loads of files a bash for loop is more convenient rather than typing out the files individually

This also works:

cat gene?.fasta > gene.fasta