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For each logged in user or for each request? I am not sure, why you want one instance per user. If you want to keep data in it, I'd rather use a cache mechanism or some other storage.

In any case, PROTOTYPE_REQUIRED is not sufficient, you will still, by default get only one instance per component.

@Component A {

@Reference(scope=ReferenceScope.PROTOTYPE_REQUIRED)
Something instance_for_A;
}

@Component B {

@Reference(scope=ReferenceScope.PROTOTYPE_REQUIRED)
Something instance_for_B;
}

While the instances in A and B are different, there are still only two of them. You either need to use a factory or some different approach. With PROTOTYPE you could do something like this:
@Reference(scope = ReferenceScope.PROTOTYPE_REQUIRED)
public void setFactory(ComponentServiceObjects<Something> factory) {
  this.factory = factory;
}
  Something x = factory.getService();
  ... do something with x
    factory.ungetService(x);

But I am not sure, how this helps you for your usecase. Maybe it does, I don't know.

I think, you should not store a component anywhere. While I tried to explain what you can do to you, I don't think that the approach is sound.
I would create a simple DTO (data transfer object) to hold the data. Store the data you need in there. When you need to do calculations, give them to the components as parameter. We use DTOs quite a lot to transfer data through the various layers.
This doesn't solve your storage problem, though. The main question is: Can you afford to lose that object at any time? If it contains only intermediate values and basically is used to speedup things, a cache is the best option. If losing these data is really bad, you should persist them in the database.
For cache, you can use the SingleVM or the MultiVM cache. Here is some Liferay code that uses it. If you have no cluster, use SingleVM, if you have a cluster, then it depends. SingleVM could still be fine for your usecase. You probably need no listeners, you just need to put and fetch objects.
https://github.com/liferay/liferay-portal/blob/7.2.x/portal-impl/src/com/liferay/portal/template/BaseTemplateResourceCache.java