Let $X$ be a $CW$-complex and $\text{Vect}^n(X)$ the collection of $n$-dimensional real vector bundles over $X$. Let $$ \text{Vect}^*(X)=\bigoplus_{n=0}^\infty \text{Vect}^n(X) $$ with addition \begin{eqnarray*} \oplus: \text{Vect}^m(X)\times\text{Vect}^n(X)&\longrightarrow& \text{Vect}^{m+n}(X),\\ (\xi,\eta)&\longmapsto&\xi\oplus\eta \end{eqnarray*} and multiplication \begin{eqnarray*} \otimes: \text{Vect}^m(X)\times\text{Vect}^n(X)&\longrightarrow& \text{Vect}^{mn}(X),\\ (\xi,\eta)&\longmapsto&\xi\otimes\eta \end{eqnarray*} where $\oplus$ denotes the Whitney sum and $\otimes$ the tensor product. Then $KO(X)$, the K-theory of $X$, can be identified with the group completion of $(\text{Vect}(X),\oplus)$.
On the other hand, by the Atiyah-Hirzebruch spectral sequence, $$ E_2^{p,q}=H^p(X;KO^q(*)) $$ converges to the generalized cohomology theory $KO(X)=\oplus_{n\in\mathbb{Z}}KO^n(X)$. The generalized cohomology theory has addition $+$ and cup-product $\smile$.
Question.
(1). What does $\text{Vect}^1(X)$ correspond to in $KO(X)$? Does $\text{Vect}^n(X)$ correspond to $KO^n(X)$?
(2). What does $\oplus$ correspond to in $KO(X)$? With the addition $\oplus$, does the monoid $(\text{Vect}^*(X),\oplus)$ isomorphic to $(KO^n(X)\mid _{n\geq 0},+)$?
(3). With the addition $\oplus$ and multiplication $\otimes$, does the monoid with multiplication $(\text{Vect}^*(X),\oplus,\otimes)$ isomorphic to $(KO^n(X)\mid _{n\geq 0},+,\smile)$ (I think it is wrong)?
No. Vector bundles all land in $KO^0(X)$.
Direct sum corresponds to addition (again, all happening in $KO^0$).
Tensor product corresponds to multiplication.