What would be the correct way to reply to a statement such as: "I belong to this group"
Would it be "So do I" or "So am I"?
You use "so am I" when you are responding to some form of the verb be:
I'm hungry. So am I.
I'm eating ice cream. So am I.
You use "so have I" when you are responding to some form of the verb have. You will always use "so have I" when have is the auxiliary verb for the present perfect:
I've eaten too much ice cream. So have I.
You may use either "so do I" or "so have I" when have is used on its own to denote possession of something:
I have an ice cream cone. So do I/so have I.
Americans in this case will pretty much always use "so do I", whereas British will use one or the other.
Other auxiliary verbs will repeat the verb used as well:
I would like some ice cream. So would I.
I should eat some ice cream. So should I.
I must eat some ice cream. So must I.
I ought to eat some ice cream. Me too!
("So ought I" sounds hopelessly stilted. You can always say "me too" instead of any of these.)
Ordinary verbs will always use "so do I":
I like ice cream. So do I.
I need to eat some ice cream. So do I.
Since the question contains the word "belong", then "So do I" is more preferable.
As an auxiliary we use do for most verbs (except not for be, will, have got and modal verbs), therefore, do is selected.
Do is generally used to take the place of a verb without auxillaries if you don't want to repeat the verb itself, typically in response to a question.
I wonder if John went to the store.
Do you want to build a snowman?
If you have a form of be as part of the verb, it's shortened to just the form of be.
I wonder if John was going to the store.
Will you be building a snowman?
No, I won't.
If you have a form of have as part of the verb, it's shortened to just the form of have.
Did you find out if John had ever gone to the store?
It turns out he had.
I didn't think I could make her do it, but she has built a snowman.
I'm surprised she has.