The first thing I noticed is the poor grammar used in the explanatory text. However, let's put that aside and address your question.
*The manager is having his problems but we have ours as well.
This doesn't work overall. In a comparison sentence such as this one, you must use the tenses and moods consistently:
*The manager is having his problems but we are having ours as well.
The manager has his problems but we have ours as well.
But is "The manager is having his problems" grammatical? Here, it starts getting complicated.
The manager is having problems.
The manager is having problems with X.
These are both fine and correct. You can replace "having" with "experiencing" without changing the meaning of the sentence.
However, "his problems" indicates that the problems are his, therefore he owns or possesses them, not that he is experiencing them.
When we add the matching clause, we must either repeat "problems" or skip "having":
The manager is having problems but we are having problems as well.
The manager is having problems but we are as well.
Compare this with:
I am having great fun now.
Whose fun am I having? This is a nonsensical question. I'm just having fun, not someone's fun.
However, as a counter example:
He is having his childhood all over again.
This is a correct way to say that he is experiencing his childhood. A childhood is a period of life that is experienced. A problem (in the sense of a difficulty) is not something specific, so his problems are constantly changing. I think that's why "having his problems" doesn't work, because the problems are not definite and fixed. They're not (all) the problems he has, they're just some problems he has.