Generally, members of a species have different DNA makeup from other species. However, as I understand it, chromosomal DNA is difficult, if not impossible, to get from ancient fossilized remains, so most paleontologists rely on structures to indicate whether speciation has occurred when such remains are analyzed.
Great question. The biological definition of a species is based on reproductive compatibility in the wild. If two organisms can naturally mate and produce fertile offspring in nature, they're considered members of the same species. This definition of species thus doesn't apply to fossil material, for which reproductive compatibility isn't known or can't be tested. So the short answer to your question is that scientists could not distinguish breeds within a species from different species just by examining similar fossils.
However, a lot is known about the range of morphological variation present within members of a living species. In general, nondomesticated species have a very limited range of morphological variation. Your dog example involves a species that has been radically altered by human breeders through artificial selection. In species of animals not subjected to artificial selection, one almost never sees as much morphological variation as we find in different dog breeds.
So scientists assign a species name to fossils by inference, considering the typical amount of variation seen within natural species and the similarities and differences to living species and preexisting known fossils. If a fossil looks significantly unlike any recorded live species or fossil species, it's usually given a new species name. However, new fossil discoveries continually improve this nomenclature. Species assignments of fossils are based on comparative evaluations of what's known when they are discovered; it's therefore common for subsequent fossil discoveries to contribute to revisions of the species names given to older fossil material.
Although DNA sequence data doesn't address the biological species definition of reproductive compatibility, it does provide a powerful data set to help evaluate species assignments. Although there is not yet a lot of DNA data from fossils, there's a lot of data on the typical DNA sequence divergence seen between members of the same species. Thus, DNA data can be used to infer species relationships. As far as the prevalence of fossil DNA evidence, recent research that used cheaper DNA-sequencing technologies successfully recovered significant portions of DNA from extinct organisms such as the cave bear and Neanderthal hominids. In the future we will have DNA data for more fossil material. Already scientists have begun to build a powerfully detailed DNA data set for studying the relationship between different fossil and living taxa, and it will undoubtedly improve our understanding of the tree of life of organisms both past and present.
For more information:
A site about cave bear fossil DNA sequencing.