What does "China’s Confidence Rises in Its Military" mean?

Solution 1:

China's confidence rises in its military.

This can be interpreted in two different ways.

The first interpretation is what you've provided:

China’s confidence in its military rises.

However, it's possible to also interpret it to mean this:

The level of confidence among China's military personnel rises.

In other words, it could be saying that Chinese military personnel have an increased level of confidence.

The second interpretation is not as likely, but there's nothing about the syntax of the sentence that rules it out.

Of course, the body of the article itself indicates that it's the first interpretation that was intended. But if you're only looking at the sentence itself, the meaning is ambiguous.

Solution 2:

Is it exactly the same as "China’s Confidence in Its Military Rises"

Yes they mean the same. There is a slight difference in emphasis. The first emphasises that China’s confidence is rising and then goes on to say why. The second allows us to wonder for a moment whether China’s confidence in its military is rising/falling or changing in some other way. As a news headline, the first is preferable because it gets the reader's attention to the important point more quickly.

Both versions are grammatical.

Grammar

"China’s Confidence Rises in Its Military" ('in its military' is adverbial to 'rise')

"China’s Confidence in Its Military Rises" ('in its military' is adjectival to 'confidence')