There are a lot of different methods to test samples for drugs. The most common for a "screening" test, like one might encounter before being hired for a job, are based on immunoassays. These use antibodies to detect certain substances in the sample. These are good for testing for a few drugs or categories of drugs (like THC, benzodiazepines, opiates, amphetamines, etc.), however they can't test for a large number of drugs at once, and often won't tell you the specific drug (e.g. it'll say "positive for benzodiazepines", not "positive for alprazolam"). If someone tests positive on one of these screening tests, then the samples can be re-submitted for further testing to clarify what is causing the positive result.
Drug testing for elite athletes is a whole different story. The World Anti-Doping Agency has established protocols for the Athlete Biological Passport, which uses standard methods to test samples and record various biological parameters over time, to ensure that athletes aren't manipulating their bodies artificially.
The exact details of how this is done is pretty complex, but they don't need to divide samples into a bunch of separate samples and test each one for a single drug. These standardized kits are able to be run on one urine sample (divided in two) and a couple of tubes of blood, the same size you'd get drawn at a doctor (just 3-5 mL each).
One way they test for many different substances using a single tube is through gas-chromatography mass-spectoscopy (GC-MS), which is a combined method to separate apart substances in the blood and identify them by their molecular weight. By comparing athlete's blood and urine samples to known patterns produced by banned substances, they're able to test for many different compounds using a single sample.