IANAB, but from what I do understand. It depends what you mean by different genes. Information wise, DNA is a string of base 4 digits(nucleotides) in groups of 3 digits, these groups are called codons. Each codon corresponds to a specific amino acid*. A protein is made up of a bunch of different amino acids chained together. The gene determines which amino acids are chained together and in what order. This long chain of amino acids tends to fold up into a complex 3 dimensional structure, and this 3 dimensional structure determines the protein's function.
Now, there are a couple ways a gene could be different without altering the protein's function. It turns out multiple codons can code for the same amino acid. So if you switch out one codon for another which codes for the same amino acid, obviously you get a chemically identical sequence and therefore the exact same protein. The other way is you switch an amino acid, but this doesn't meaningfully affect the folded 3D structure of the finished protein, at least not in a way that alters its function. Both these types of mutations are quite common; because they don't affect function, they're not "weeded out" by evolution and tend to accumulate over evolutionary time.
* except for a few that are known as start and stop codons. They delineate the start and end of a gene.
You could build houses from bricks, timber or poured concrete that all looked the same in the end. Their internal structures and methods of construction would be different, but they would have the same form.
For a given output, you could write a program in wildly different programming languages, or even use the same language but structure it in wildly different ways.
If there's no match for the source code (genes), then find a match for the output (protein).
In terms of 3D fold - i.e. the general abstract shape of the protein in 3D, you can make loads of substitutions without changing it, generally as long as you stay within the same class.
It's not until you compare the 3D shape that you see the relationship.
