Does $E[e^{it(aX + bY)}]=E[e^{itaX}]E[e^{itbY}]$ for every $a,b\in\mathbb{R}$ imply that $X$ and $Y$ are independent? Let $X, Y$ be two random variables such that for every $\alpha, \beta \in \mathbb{R}$, 
$$E[e^{it(\alpha X + \beta Y)}]=E[e^{it\alpha X}]E[e^{it\beta Y}]$$ for all $t\in\mathbb{R}$.  Does it follow that $X$ and $Y$ are independent?
 A: The answer is YES. Let me introduce some notations first.
Suppose $X=(X_1,\cdots,X_n)$ is an $\mathbb{R}^n$-valued random variable (i.e., a random vector). The characteristic function for $X$,  denotes as $\varphi_X(u)$, is a function from $\mathbb{R}^n$ to $\mathbb{R}$: 
$$\varphi_X(u):=E(e^{iu\cdot X}),\quad u\in\mathbb{R}^n$$
where $u\cdot X$ is the inner product in Euclidean space. In particular, when $n=1$, we have the characteristic function for random variables. Note also that
$$
\varphi_X(u)=\varphi_{u\cdot X}(1)\tag{1}
$$
The answer to your question follows directly from the following theorem (Probability Essentials by Jacod and Protter, Chapter 14):


To see how your question fits in this setting, let $Z=(X,Y)$ (these $X$ and $Y$ are random variables in OP) and $t=1$. Let $u=(\alpha,\beta)\in\mathbb{R}^2$. Then 
$$
E(e^{iZ\cdot u})=E(e^{i\alpha X})E(e^{i\beta Y}).
$$

[Added due to the confusion in comments]
Note that one can interpret the question in OP in the language of characteristic functions as the following way:  

If 
  $$
\color{blue}{\forall (a,b)\in\mathbb{R}^2\ \forall t\in\mathbb{R}}\ \quad \varphi_{(a,b)\cdot (X,Y)}(t)=\varphi_{aX}(t)\varphi_{bY}(t)\tag{2}
$$
  then do we have that $X$ and $Y$ are independent?

Note carefully that the condition $(2)$ implies in particular that
$$
\color{blue}{\forall (a,b)\in\mathbb{R}^2} \quad \varphi_{(a,b)\cdot (X,Y)}(1)=\varphi_{aX}(1)\varphi_{bY}(1)\tag{3}
$$
which, by $(1)$, is equivalent to say that 
$$
\color{blue}{\forall (a,b)\in\mathbb{R}^2}\quad \varphi_{(X,Y)}(a,b)=\varphi_X(a)\varphi_Y(b).\tag{4}
$$
In order to apply the theorem, note that $a,b$ here play the role of those $u_i$'s.
Note also that in $(2)$, $\varphi_{(a,b)\cdot(X,Y)}(t)$ is the characteristic function of the random variable $aX+bY$ when $a,b$ are fixed (and $t$ is the variable of the characteristic function). While in $(4)$, $\varphi_{(X,Y)}(a,b)$ is the characteristic function of the random vector $(X,Y)$ evaluated at $(a,b)$. $(1)$ tells you the relationship between these two objects.
