Yes, they're synonyms.
Or else is two words, so it’s not a part of speech. Or is a conjunction, which introduces a coordinate phrase or clause, meaning that the next phrase/clause is an alternative to the previous one. Else is an adjective or adverb, which means “in addition to what has been covered or seen so far.” Though its meaning is the same as otherwise, else is much more limited syntactically (see below).
Otherwise can serve as adverb or as a combined conjunction and adverb, meaning the same as or else. Otherwise usually doesn’t serve as an adjective, though I suppose you could force it into that role.
(Of course, since this is English, a very phrase-oriented language, I suppose you could say that or else is a phrasal conjunction. Most grammarians would disagree, and I’m hiding this remark inside parentheses, but I think it’s a reasonable idea.)
Examples of else
Is there someone else up there we could talk to? [Source. The speaker was displeased with the person who greeted him at a castle, so he asked to talk to someone else.]
There, else is an adjective modifying someone.
You must find her attractive, or else you wouldn’t be talking about her so much.
There, else is an adverb modifying the clause you wouldn’t be talking about her so much.
You really don’t need else after or, because or means that the upcoming phrase or clause is an additional alternative to what has come before:
You must find her attractive or you wouldn’t be talking about her so much.
The reason for saying or else instead of just or is partly to get the emphasis that comes from two words instead of one, sometimes maybe to get a nicer rhythm, but mainly because English is somewhat phrase-oriented (as I was saying in parentheses above) and listeners tend to perceive or else as a single unit of meaning (which would make it a two-word conjunction, but this is a decidedly nonstandard analysis).
You can also substitute else for or else, making else serve as a conjunction, though I think this is mostly heard in British English:
You must fancy her, else you wouldn’t be talking so much about her.
Examples of otherwise
You can substitute otherwise in the previous example:
You must fancy her, otherwise you wouldn’t be talking so much about her.
Otherwise can fit into many places where else won’t fit:
Otherwise, the roof will collapse.
You’ll have no alternative otherwise.
Only a fool would do otherwise.
You could not substitute else for otherwise in those sentences, at least not in American English. Else as an adjective usually appears after the noun it modifies, and it’s usually a stock phrase like someone else, anyone else, something else, anything else, what else, how else, or else, etc. (Did I mention that English is somewhat phrase-oriented?)
Or else also has an idiomatic meaning when it’s not followed by a clause, like this:
Pay up…or else.
This is what a gangster says to threaten someone with violence if they don’t do as the first part of the sentence says. It’s short for something like “Pay up, or else there will be consequences that you will find very unpleasant.” This usage of or else is somewhat ominous because the exact nature of the threat is unspecified.