Learn Italian: Hanno comprato la medesima macchina.

When medesimo (or stesso) comes before the noun, it means "the same" (identity). If you placed it after the noun (la macchina medesima), it would mean "the car itself" (reflexive emphasis). Saying "They bought the car itself" wouldn't make much sense in this context.

Could I use stessa instead of medesima in this sentence?

Absolutely. Hanno comprato la stessa macchina means the exact same thing. Medesimo is just a slightly more formal, elevated synonym for stesso. You will hear stesso much more often in everyday conversation, but it is important to recognize medesimo.

Since medesima acts as an adjective here, does it always need a definite article like la?

Yes. When meaning "the same," both medesimo and stesso are almost always preceded by a definite article (il, la, i, le) or a demonstrative (like questo or quello). This is because you are pointing out a specific, identical thing.

Does this sentence mean they bought one shared car, or two identical cars?

Just like the English phrase "they bought the same car," it is technically ambiguous. Most of the time, this phrasing implies they each bought the exact same model of car. If they literally pitched in to buy one single vehicle to share, context would usually make that clear.