Hi, and welcome to game dev!

First of all, it's great that you have someone supportive in your life. It can be very hard when the ideas are just stuck in your head and you can't act on them or talk about them in the way you want to. I often turn to writing in my personal diary now when I need to reflect on things and make big decisions and I'm feeling stuck. If you aren't already, it's a good habit to get into.

Regarding the game idea, there are a bunch of motivations to disentangle there:

• Raising money for rescue centers
• A game scenario revolving around dog rescues
• Gameplay based on a genre concept (shooter game)
• A specific feature of the game software(real stories, in-game catalogue of these stories)

These are all basically good ideas, when we put them in a list. You can see and imagine a way to execute on each one. Do they all go together? That part is more tricky.  The game's scenario, its reason for being, and the catalogue feature all make sense. But as you point out yourself, there is some kind of gap between "shooter game" and "dog rescue scenarios" that we have to tackle by working on the game design some more. So let's focus on that.

Here is a technique I use which is modeled on the classical "theories of truth" that appear in introductory philosophy courses. Read up on those first to get yourself oriented as to how I'm using them.

First, establish some ground rules about what the design goals are: What do we expect the player to do and to feel, and what methods are we implementing to convey that? What does the game believe is "true" about the world? (e.g.  "it is good to rescue dogs in danger" or "a violent intervention like shooting is OK") Brainstorm, and then try to make the things the game believes are true be justified by the things the player is seeing and doing: for example, "the player earns points for each dog rescued" justifies our belief that rescuing dogs is good, and "there is a character shown immediately threatening each dog" justifies the use of violence against that character. What we're ultimately looking for in the implementation part of things is a close correspondence to reality - it just makes sense to have those specific ideas and rules in the game, and where there are gaps like "it wouldn't work like that in the real world" we can lean on pragmatic justifications. 

You have a great starting point here because you can do this initial brainstorm by researching real rescue stories and then also listing things that already exist in shooter games. It's much easier to borrow and reuse elements than to try to develop them "from scratch".

After you've brainstormed a bit, you might find that while you have very specific elements that are cool ideas, like the catalogue feature, the game as a whole is still not coherent yet - the implementation is not succeeding at justifying the beliefs, nor do the beliefs motivate the implementation, and it's drifting away from being a specific genre like "shooter" too because there are too many competing elements. At this point it's time to try to filter things down and revise them into a smaller, more prioritized number of ideas that can cohere better: If a very specific belief like "shooting is OK" doesn't seem to be working, maybe a different common-sense phrase like "you should help those that can't defend themselves" will make it fit, and it motivates some different angle for the implementation like "the player has an ability that prevents dogs from getting into danger".  Some very big breakthroughs in your ideas can happen by revising towards better coherence and more efficient use of a limited "idea budget". You may have to make a hard decision about what to keep or cut, and it's much better to make that decision before you've started building anything!

After doing enough revision, eventually it will become very hard to make things cohere any better, like screwing in a bolt tighter and tighter until it just won't turn any more.  At that point you can show it around for more feedback, or move on to thinking about how to execute on the idea!