What does it mean to share similarities among living organisms "both horizontally and vertically"?
Ben Bolker's comment is correct; the next sentence in the passage helps to clarify (bold added):


  The biggest spin off
  of such studies was the recognition of the sharing of similarities among
  living organisms both horizontally and vertically. That all present day
  living organisms are related to each other and also to all organisms
  that ever lived on this earth, was a revelation which humbled man and
  led to cultural movements for conservation of biodiversity.


The relationship between species in the present day are the horizontal relationships; one can define species horizontally by looking at and grouping organisms existing today (or at any other time "slice"). You can compare species horizontally by, say, looking at the skull of a human and the skull of a chimpanzee.

The relationship between species today and species that have existed in the past are the vertical relationships. For example, you could look at the skull of a modern horse and compare it to the skull of Mesohippus.

I've included a couple sources below that use this distinction, though I'm not sure where it originates.

I'd say the "horizontal" terminology comes up more frequently and may be more familiar, for example in reference to horizontal gene transfer one is talking about genetic material moving between species living at the same time. "Vertical gene transfer" can also be defined as the transmission of genetic information from parent to offspring, but more typically you would just call it inheritance.



Alvarez-Venegas, R., Sadder, M., Tikhonov, A., &amp; Avramova, Z. (2006). Origin of the bacterial SET domain genes: vertical or horizontal?. Molecular biology and evolution, 24(2), 482-497.

Stamos, D. N. (2002). Species, languages, and the horizontal/vertical distinction. Biology and Philosophy, 17(2), 171-198.