Do all of these mean just "fat, obese"? Dictionaries seem to indicate that these characters mean that separately. Is there a reason to use them both? Is this one of the cases where two characters just remove ambiguity?
Explaining from the angle of Mandarin,
肥胖 -> it is a noun form meaning obesity. Obesity is a specific term, and is referring to general public, so you should not use it simply to say someone is fat.
胖 is used with people in other situations.
肥: I don't think it is used with people in Mandarin, but only for other objects and usually used to make compound words only. Ex: 肥牛(fat cow), 肥皂(soap)
FYI, 肥 is still used with people, but not in Mandarin, mostly in Cantonese speech.
Is this one of the cases where two characters just remove ambiguity?
Yes, Since each Chinese character has its own meaning or meanings. A single character, usually cannot clearly express a specific meaning, therefore two characters would combine into a specific word, and two words may combine into one specific phase.(see my other post How to determine if character can be used separetely?)
In the case of 肥 and 胖. both can be an adjective for "fat", however 肥 also has other meanings:
 [adj] fat; plump; chubby | [antonym] 瘦
 [adj] greasy; oily; fatty
 [adj] fertile; rich; luxuriant
 [n] fertilizer; manure
 [adj] profitable; lucrative; affluent
Combine 肥 and 胖 into 肥胖 make it a specific term for "fat /obese ". Since it is a specific term, it is also more formal.
Beside being an adjective for "fat", 肥胖 can also be a noun for "obesity"
They have different "strength" and meaning:
胖. Yes, they have similar meaning but cannot be used randomly. Mostly
肥胖 is kind of medical situation, which means a serious illness.
肥 is very humiliating. Although
胖 is less aggressive, it is a little rude anyway.
It's easy to distinguish them:
肥 is more appropriate to describe animals like "好肥的羊/牛/猪"
胖 is more appropriate to describe human like "你/他/她 好胖"
Former two are adjective While 肥胖 it's a noun,it's a status, usually be used alone like "肥胖的原因是…"/"…导致了肥胖"
In addition to the very insightful answers so far, I would like to quote a fragment from Keat’s Chinese:
In our daily life, we may sometimes want to describe somebody is fat. So when if you learn Chinese language in China, you should know the two different words expressing fat “胖” and “肥”, both are opposite to “thin”. The difference is that we use them to different objects. “胖” is often used to describe the person such as “身体发胖” and “长得胖”. On the contrary, “肥” is to applied to animals or the clothes or shoes. So we can say “肥肉” and “肥猪” as well as “裤子有点儿肥”. So it is very impolite if we say somebody is “肥”.