Say that we have two i.i.d random variables $X1$ and $X2$ which takes values +1 and -1 with probability 1/2 respectively. Define $Y=X1+X2$ and $Z=X1-X2$. Then we can show that both $Y$ and $Z$ takes values -2,0,+2 with probabilities 0.25, 0.5, and 0.25 respectively.
My question is: since both $Y$ and $Z$ have same distribution, can we say that they are 'dependent'? More generally, if two discrete random variables have the same distribution, are they necessarily dependent?
The notions "two random variables have the same distribution" and "two random variables are dependent" are completely distinct.
Same distribution and independent: the result of two coin flips are independent of each other (knowledge of the result of the first flip does not give you any information about the second flip), but have the same distribution (half heads, half tails).
Same distribution and dependent: in your example, $Y$ and $Z$ are dependent, because information about $Y$ can give you information about $Z$ (if you know $Y=2$, then you know $Z=0$).
Different distribution and dependent: in your example, $Y$ and $X_1$ are dependent (if you know $Y=2$, then you know $X_1=1$) but have different distributions.
Different distribution and independent: a coin flip and a roll of a 6-sided die