How can a particle's position and velocity be the same all the time? I have an $x-t$ graph (position-time) which its equation is $x=e^t$.
If we get derivative of it, the velocity's equation will be $e^t$ too.
How can a particle's position and velocity be the same? what does it mean when position and velocity are the same? is there any real examples of it?
 A: Remember that these quantities have units. What does $\exp\left(1\,\mathrm{s}\right)$ mean? How does it compare to $\exp\left(1\,\mathrm{min}\right)$?
Generally you can only take non-polynomial functions* of dimensionless quantities. So for your expression to make sense it should be something like 
$$
x = x_0 e^{t/t_0}
$$
Now it may be that $x_0 = 1\,\mathrm{m}$ and $t_0 = 1\,\mathrm{s}$, in which case, in SI units, it will numerically look like $x=e^t$, but if we change units to say miles and days this will no longer be true.
If we take the derivative of this equation we find that 
$$
 v = \frac{x_0}{t_0}e^{t/t_0}
$$
which is clearly not the same as the previous equation,(although again, if we make a particular choice of units it may look the same).
*logarithms are a bit of an odd case here. If I have an expression like $\log t/t_0$ then I can (formally at least) use rules of logarithms to write this as $\log t - \log t_0$. This appears to the logarithm of a dimensionful quantity, but if you make a change of units you will find that numerically everything cancels out. Generally I find it is safer and makes more sense to simply say that the logarithm identities only apply to dimensionless quantities and so disallow the second expression, but some people use other conventions. 
A: There is no physical significance of the equations velocity and position w.r.t time being same simultaneously but what we can understand from this is that both phenomenon change simultaneously
It is possible for position and velocity to be same but both of them depend on the frame of reference at any point of time the position and velocity are same w.r.t. independent frames or the same frame as it is in this case 
Hope you got that!
