"The problem is he is stingy"
Solution 1:
It's perfectly grammatically correct.
The problem is just that: you repeat the word "is".
It's that simple.
It's a commonplace in English, say speechwriting or advertising writing (I mean say for radio or TV voiceovers), that you don't repeat a word in a sentence or, really, in a passage, and particularly not close.
Here you have two "is"s very close, so it sounds awkward. It's that simple.
(AND INDEED, he/his are almost the same word - that's another repeat.)
The solution is this simple:
The problem is this: he's very stingy with money.
The problem is: he's very stingy with money.
or any other combo.
{Note - it goes without saying there are many other situations where, for emphasis, for drama, because you are Winston Churchill, etc, you repeat a word closely. Say ... "May the luck be with the player." .. repeating the "the" sounds cool and dramatic there. But in the example at hand, it's simple: the sentence sounds lousy since you have (indeed two different) close repeats.}
Solution 2:
Not sure how correct it is right now; is it just me or does it need a comma? Like so: "The problem is, he is very stingy with his money."
I know for sure that "The problem is that he is very stingy with his money." is correct though.