You have your own particular, specific problems — financial, intimate, psychological, and otherwise.
"otherwise" is usually put to make a contract, to show that something is different from what was previously mentioned. In this sentence, though, it looks like it's conveying the meaning of "and so on"/"and so forth"/"and the like".
I've never encountered this usage before. It seems strangely unfamiliar, and it doesn't fit into my understanding of the word.
The quote is from the book 12 Rules for Life: An Antidote to Chaos by John Peterson. Here is the full context:
Be cautious when comparing yourself to others. You're a singular being, once you're an adult. You have your own particular, specific problems — financial, intimate, psychological, and otherwise. Those are embedded in the unique broader context of your existence.