There are a lot of good synonyms for identify—which one to use depends on what exactly you want to mean. Here are some of the main ones:

• recognize—to be able to tell what something is (and often to know its name), as in He immediately identified/recognized it as a monarch butterfly. 
• diagnose or pinpoint, as in My mechanic was finally able to identify/diagnose/pinpoint the problem. 
• associate, such as with a certain group or belief, as in She’s known for identifying/associating with the party’s reform wing. 
• distinguish, as in Her height identified/distinguished her as the volleyball player. 
• establish (who someone is), as in He has been identified/established as the main suspect. Relatedly, to identify someone is to figure out who they are—to establish their identity.

Sometimes, to identify in some way means to consider some characteristic as an inherent part of your identity (and perhaps to make this publicly known), as in I identify as nonbinary or She proudly identifies as Mexican American. Another way to express this is simply by using forms of the verb be, as in I am nonbinary.

Yes, identify and determine can be synonyms when they mean the same thing as diagnose and pinpoint, among other things.

Let’s say you’re taking a test that asks you to identify the world’s oceans. In this case, yes, you’re required to list them. Of course, if you’re asked to come to the front of the classroom and identify them on a map, you’re really being asked to demonstrate that you can recognize them by pointing them out (as opposed to listing them out loud or on paper).

There are a few different words that you can use as the opposite of identify.

The most direct opposite of identify is misidentify, which means to identify something incorrectly. For example, a newspaper story might misidentify someone by providing the wrong name. The police might misidentify a suspect—meaning they got the wrong person. A birder might misidentify a bird due to thinking it’s a different one that’s similar.

When identify means to determine, diagnose, or pinpoint something, the opposite of doing this is to overlook or miss something entirely. To misdiagnose a problem is to incorrectly identify something as the problem (when it’s really not).