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Dirt is defined as any unclean substance, such as mud, dust, excrement, etc; filth. However, I am not sure if dried pee is considered dirt, because it's not visible and it's barely solid. You can't scrub it off and it's soluble. So is it considered dirt or not? I know this question is weird and sound stupid, but the answer isn't as simple as it might appear.

For example:

He went to the bathroom to pee a while ago, so he took a bath to clean
off the dirt on his private parts.

'Dirt' in the sense of bodily contamination, is usually reserved for visible matter including, as you say, soil, excrement, dust, soot, etc. Cambridge gives a general definition:

dust, soil, or any substance that makes a surface not clean

and a specifically American one:

earth or a substance like it that has gotten on the surface of
something such as your skin

I would say that most British people would accept the latter meaning, but, if pressed, concede that dried urine is a 'substance that makes a surface (including the skin) not clean'. I would add, although perhaps I am being subjective here, that most people would not feel the need to take a bath after urination. Males who are old enough to use a toilet can generally manage not to get urine all over their private parts.

I wonder if the question is about ritual cleanliness?

"Dirt" has a range of meanings, from ordinary soil that plants grow in, to "a substance that worsens the appearance of something", to metaphorical meanings.

Surely "dried pee" does worsen the appearance of something, if not by sight then at least by smell. So dried pee could be "dirt".

In the quote given, I find the use of "dirt" to be odd. Normally going to the bathroom, you don't get pee on your private parts (it comes out but goes into the toilet). If you had said,

The boy wet his bed, so had to wash the dirt out of his sheets

I would understand. However I can't see how the sentence is clearer than

The boy wet his bed, so had to wash his sheets

He went to the bathroom a while ago, so took a bath to clean his privates.

I see no benefit in mentioning "dirt" in the particular example sentences.

This is English SE, not Chemistry. In the English language the adjective "dirty" has a very expansive meaning, usually nothing to do with with soil. Like "dirty dancing" for example. The quotation you provided is super weird, personally I would be very surprised if it were from an authoritative writer.
On the other hand, dried shit generally does not smell bad, unlike dried pee. Thus in many Asian and African cultures, people cook on a fire made with dried cowshit.
Thus is is more accurate to think about "dirt" and "dirty" as having to do with using the private parts, and not so much with substances they produce, haha.