When catching Pokemon, we all see the pokeball shake back and forth. Does more shakes actually indicate that I was close? Is there any useful information I can get from the number of shakes?
Like in the rest of Pokemon video games, it shakes up to 3 times when you catch it.
Each throw has an individual chance. If you throw and 2 shakes occur but the Pokemon escapes, that has no effect on your next throw -- you might catch it, or it might shake once and escape.
In other words, I don't consider you to gain any useful information about it at all.
It's in all games and just a design choice to build tension and excitement. Otherwise this would complicate game mechanics without any real benefit. There's no difference in luck no matter whether it escapes immediately or after shaking two times.
You basically roll the dice once: Do you catch the Pokemon or do you fail?
If you catch it, everything is done.
If you don't catch it, a second roll determines how many shakes are actually shown.