Complete graph invariant
The  answer is no. A counterexample is given by the following two graphs both having $(-1,
 -1,
 -1,
 -1,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 0,
 1)
$ as the multiset that you describe.





If you want to play further, you can use the following Sage program that found them

def tupleDet(G):
    A = G.am()
    L =  []  
    for sub in subsets(range(G.order())):    
        D = A.matrix_from_rows_and_columns(sub,sub)
        L.append(D.det())

    return tuple(sorted(L))    


def search(n):
    d= {}
    for G in graphs.nauty_geng(str(n)):
        des = tupleDet(G)
        if des in d:
            print G.graph6_string(), d[des]
            return
        else:
            d[des] = G.graph6_string()