1

When I first saw these sentences bellow, I thought they meant "Have someone +adj around yourself",

You have me sad for most days

They'll have you suicidal

But, when I translated, I saw that it isn't like that, but more like: "You make someone +adj"

Now I am confused. What do those sentence actually mean?

2

Yes, that usage overlaps with "make someone" +adj. Here is a dictionary definition that includes that meaning:
Oxford Lexico "have" 2.4

2.4 Cause (someone or something) to be in a particular state or condition.
Look at the additional example sentences, in particular, this one:
‘Now her article has me thinking.’

This applies to your second example sentence. Your first example sentence doesn't sound quite right, so unless you can point to a source, I won't comment on it.

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