This explains it better than the actual news article.
The popular factoid is correct, but the confusion here is that these are different measurements. Humans and chimps' genomes are similar in that if you align all the bases [A,C,T,G] that can be unambiguously aligned between the two genomes, 98.8% of the bases are identical. For modern humans to Neanderthals, that number is 99.7%, and between two random modern humans, it would be ~99.9% on average.
This paper is asking a subtly different question - how much of the modern human genome is strictly human, not by simply lining up bases and running a diff, but looking at the inheritance of chunks of DNA ("haplotype blocks", size determined by processes of recombination, etc.) to try to understand how much and which regions of the modern human genome came from interbreeding with Neanderthals or Denisovans. There was variation in the pre-human population before the human/Neanderthal split, which means that if you compare just a single human to a single neanderthal, you'll find unique variants to each. However, most of those variants will have existed in both the human and neanderthal populations, so they should count neither as uniquely human nor neanderthal (knows as Incomplete Lineage Sorting, or ILS).
The chunks in modern humans that derive from Neanderthals or Denisovans are different in different people and broadly across population groups (e.g. highest percent introgressed in Melanesians, lowest in Africans). But across all the modern humans in the study, there are regions where Neanderthal/Denisovan inheritance or shared variation (ILS) was never seen - that's 7% of the genome ("deserts"). And just 1.5% of the genome was in chunks where moderns human commonly have a unique mutation compared to Denisovans/Neanderthals.