This is, ultimately, a question of law and legislation. Intellectual property law enables persons and companies to hold rights over things they invent and create. However, things that are discovered cannot be patented or owned as intellectual property. For example, scientific discoveries cannot be owned. Otherwise Peter Higgs could be the owner of his boson. NASA might have to get a licence from the Estate of Sir Isaac Newton every time they launched a satellite. These examples are a bit silly, but they are for the purposes of illustration.
The real question is whether finding genes is a matter of discovery or of invention. if a company had invented a gene, perhaps building a completely new of a kind that had never existed before in nature, say, by using the techniques of synthetic biology, then they might be able to make a good case for owning it. However, if the same gene as discovered existing in nature, then their case for ownership is weakened and could be liable for a challenge in court.
A recent challenge over the ownership of a medical gene test kit was found in favour of the gene in question being a discovery and not an invention. This resulted in the test kit being able to be used widely without having to either get a licence to do so or have to pay the owners. Since the test was for a type of cancer there were a number of cancer patients who wanted the test kit being made readily available. So they felt it was in their interests to free the test kit from commercial restrictions.
On a similar topic, the ownership rights to oil and minerals in land you own aren't vested in the land owner. If a mining company takes out mining rights to pluvidium ore that might under your property, once they find the ore there the mining company owns the mineral and has the right to mine it. You may have been the land owner, but you never were the owner of any oil or minerals under it.
Similar arguments have been applied to the genes in your body. They may be there, and part of your flesh, but that fact doesn't give you ownership of them. If it did, then possibly your parents (and even their parents) might also have a claim on the ownership of your genes.
It is remarkable that Monsanto has a legal claim over the ownership of its genetically modified seeds. The genes involved in engineering the seeds would have been found through a process of discovery. Possibly, their ownership is vested in the art they exercised in creating the modified seeds. It is also possible that the court decisions that enabled them to claim ownership over their GMO seeds were based upon a misunderstanding about how science is conducted and the judges in question may have been dazzled by the genetic engineering techniques themselves.
There are no absolute criteria to determine who knows genes. It is a matter of legal precedent and practise, relevant intellectual property legislation, and the accumulated body of case law and court decisions. Ownership of genes will be refined and decided as more genes are discovered and more claims are made about whether they can be owned or not.
People make the laws, ownership is only a matter of law. When people change their minds about how something can or should be done under the law, then the laws change too.
Pluvidium is a purely fictious mineral. Anyone who thinks they have discovered it or who claims ownership of it, needs to brought back to reality.