In difference-in-difference approach, would it be possible to add a dummy variable indicating whether a data point belong to treatment group? Since there is the possibility that there are some factors result in whether a data point belong to treatment group, I think we should add a dummy variable indicating  whether a data point belong to treatment group when using difference-in-difference approach.

However, this was not done in research generally, e.g. this paper:
Chhaochharia, Vidhi, and Yaniv Grinstein. "CEO compensation and board
structure." The Journal of Finance 64.1 (2009): 231-261.

Are there reason for not adding the dummy variable? Would colinearity be an issue? (I don't think it is an issue in the above example.)
It depends on the estimated model. Let me use the example provided in Mostly Harmless Econmetrics. 

The experiment

Suppose we are interested in the effect of the minimum wage on employment. Card and Krueger (1994) use a dramatic change in the New Jersey state minimum wage. On April 1, 1992, New Jersey raised the state minimum from \$4.25 to \$5.05. Card and Krueger collected data on employment at fast food restaurants in New Jersey in February 1992 and again in November 1992. Fast food restaurants are big minimum-wage employers. They also collected data from the same type of restaurants in eastern Pennsylvania, just across the Delaware river. The minimum wage in Pennsylvania stayed at \$4.25 throughout this period. 

Using their data, they computed diferences-in-diferences estimates of the effects of the New Jersey minimum wage increase. That is, they compared the change in employment in New Jersey to the change in employment in Pennsylvania around the time New Jersey raised its minimum.

The Fixed Effects Model

Differences-in-differences (DD) is a version of fixed-effects estimation using aggregate data. To see this, let


$y_{1ist} =$ fast food employment at restaurant i and period t if there is a high state minimum wage. 
$y_{0ist} =$ fast food employment at restaurant i and period t if there is a low state minimum wage.


Let $D_{st}$ be a dummy for high-minimum-wage states, where states are index by $s$ and observed in period $t$. In the absence of a minimum wage change, employment is determined by the sum of a time-invariant state effect ($\gamma_s$) and a year effect ($\lambda_t$) that is common across states. Assuming that $E[y_{1ist} - y_{0ist}|s,t)$ is a constant, denoted $\delta$, we have:

$$Y_{ist}= \gamma_s + \lambda_t + \delta D_{st} + \epsilon_{ist} ~~~~~~ (1) $$
(I corrected $\lambda_s$ here.)

The Regression Model

We can use regression to estimate equations like (1). Let $NJ_s$ be a dummy for restaurants in New Jersey and $d_t$ be a time-dummy that switches on for observations obtained in November (i.e., after the minimum wage change). Then

$$Y_{ist}= \alpha + \gamma NJ_s + \lambda d_t + \gamma(NJ_s \times d_t) + \epsilon_{ist} ~~~~~~ (2) $$

is the same as (1) where $NJ_s \times d_t = D_{st}$. 

Your Question

Regarding your question, a key difference is that equation (2) includes the dummy variable indicating whether a data point belongs to treatment group ($NJ_s$ the dummy for restaurants in New Jersey). This dummy is not present in model (1) but the two models are equivalent.