One way would be to simply look at the blood underneath a microscope. If you take a smear of the blood and then stain it with various dyes, blood cells would look different between different species.

The differences would be more extreme the less-related the species. For example, the red blood cells of reptiles and birds still have a nucleus and look like oval eyes. Human red cells, on the other hand, are just red circles-- slightly lighter in the middle than the sides.

Amongst mammals the differences might be a bit more subtle. For example, you might look for a specific type of white blood cell called a eosinophil. They aren't common but if you search the slide you'll probably find one eventually. Eosinophils have a lot of large granules in them that look pretty different from animal to animal.

There are, of course, many many other ways to identify the blood. Some are much more precise than this (genotyping being the obvious one.) But a peripheral smear is low-tech, inexpensive, and I think most fun.