what is wrong, by the way, with saying that we are 99% genetically similar? we frequently see statements along the lines that we humans share x% of our DNA with chimps, y% with lettuce, etc
There's nothing so much wrong with it as it being sort of a confounding question when, as you note, we also share a massive suite of DNA with other species, largely in accordance with how far removed we are from a common ancestor (although it's not 1:1).
The bits that are specific to humans, or in which variation impacts humans in particular, would be a better place to start from when considering the question, because that's where there's variation
among humans, which then makes sense of a question that asks that conditionally!
I shouldn't have toked before writing that sentence!
a recessive gene can be anywhere along the genome
Correct, but there's also the element of purifying selection Fenrir raised; there's no real general statement that could be made here, but you can imagine that some of the most ancient shared genes have undergone many, many rounds through many species, and their effect was so harmful as to have effectively made themselves extinct in the population and all daughter populations. Mutations can still produce this harmful allele, but the chances of it finding itself in a body with another copy is extremely low.
Other genes, often more recent in evolutionary terms, or those with more complicated effects such as conferring some benefit, may have more copies in the population to encounter and thus are more likely to persist in the population and... feedback loop. This is magnified by geographical proximity when individuals continually breed with neighboring relations who are more likely to have that gene; it ratchets up quickly under certain conditions, and inbreeding can quickly become endemic to small populations as there's literally no one to breed with absent those genes.
All species have to manage this in some way, and there's quite a lot of interesting evidence about our earliest modern human ancestors practicing female exogamy, or marrying daughters out to neighboring tribes.