To begin with I will give the meaning of the word 'предмет' in Russian 1) A thing, an object. (This is the most common). 2) In philosophy — what the subject's thought or action is directed at. (subject-object relationship) 3) In education (science) - the same as a subject (academic discipline) 4) the topic of something. For example, the topic of conversation. Предмет разговора. You are confused because in Russian the topic of conversation, academic discipline and so on is the object (предмет). And our actions and thoughts are directed at it. In English, it is the opposite.

'Subject and object are literally opposites.' They are opposites in philosophy, and are translated as субъект and объект here. But we don't often have philosophical conversations, so we don't use these words in their philosophical meanings. They are opposite in English sentence as two different parts if it. But in Russian we have another parts in the sentence, that don't refer to a subject or an object. So, no opposition here too in Russian.

Предмет can mean: - a piece. "Предмет мебели" is a piece of furniture, "предмет туалета" is a piece of clothing. - an item - an object, a thing. - a school subject - a subject of duscussion (предмет обсуждения). - ....of [attention, desire, interest etc.] It is a bit strange for me, that in English "object of study" but "subject of discussion". From Russian perspective, it is what you discuss/study/desire/project your interest at. "Объект" - object, as in НЛО (UFO) - a part of subject-object pair. An experiencer/doer, and the patient. - in some newer phrases like "object of discrimination" you will find "объект" instead of "предмет". They are the same word. Предмет is a Russian translation of Latin word. Объект is the same word borrowed in its Latin form. Sometimes you can use them interchangeably: объект of interest/study is only different from предмет of interest/study in style. If you use предмет in UFO, people will think it is a .... fork, for example:) In meanings "piece", "subject of discussion" and "school subject" only "предмет" can be used. Grammatical subject is "подлежащее", and object is "дополнение" (complement).

Объект - понятие более широкое, чем предмет. Предмет - узкое понятие. Например, наука микробиология. Объект - микроорганизмы. Предмет - рост и развитие микроорганизмов/ дыхание/ строение. Или например бизнес. Объект - предприятие. Предмет - финансовое состояние этого предприятия/ штат и т.д.