You can do this simply enough in awk:

$ awk -F'_' '{
                split(FILENAME,n,/\./); 
                if(/^>/){
                    sample=$1; 
                    sub(/>/,"",sample); 
                    $0=">"n[1]
                }
                print > sample".fasta"
            }' gene1.fasta gene2.fasta ... geneN.fasta

You can do this simply enough in awk:

$ awk -F'_' '{
               split(FILENAME,n,/\./); 
               if(/^>/){
                  sample=$1; 
                  sub(/>/,"",sample); 
                  $0=">"n[1]
               }
               print > sample".fasta"
             }' gene1.fasta gene2.fasta ... geneN.fasta

If I run this on the two files you gave, I get:

$ cat sample1.fasta 
>gene1
GATCTCGTAGATGTGA
>gene2
ATGCTGTGAAGTA

$ cat sample2.fasta 
>gene1
GATCTCGTAGATGTGA
>gene2
ATGCTGTGAAGTA

Explanation

• -F'_' : set the field separator to _ as an easy hack that lets us use $1 for the sample name (sample_NNN, so sample will be $1 and NNN will be $2).
• split(FILENAME,n,/\./); : FILENAME always holds the name of the file currently being processed. Here, we are splitting it on . to get the gene's name. This assumes all of your files are named $geneName.fasta. The split creates the array n whose 1st element, n[1] is the gene name.
• if(/^>/){ : if this is a fasta header line.
• sample=$1 : set the variable sample to the sample's name. See the first note about -F'_'.
• sub(/>/,"",sample); : remove the > from the sample variable. We don't want that in the file name.
• $0=">"n[1] : set the current line to be > and the gene name.
• print > sample".fasta" : print each (modified) line into the appropriate file name.