The wikipedia gives a good example:
Two variables are perfectly collinear if there is an exact linear relationship between
the two. For example, $X_1$ and $X_2$ are perfectly collinear if there exist parameters
$\lambda_0$ and $\lambda_1$ such that, for all observations $i$, we have
This applies directly to time series, just change $i$ to $t$.
Two times series $X_1$ and $X_2$ are cointegrated of order 1 if they are
Integrated of order 1, meaning that first differences of $X_1$ and $X_2$ are stationary processes.
There exists parameters $\alpha_1$ and $\alpha_2$ such that linear combination
is a stationary process.
So the answer to your first question is no.
The answer to your second question can be yes. The colinearity is in a sense stronger, i.e. more restricting property. From statistical point of view colinear time series are the same, i.e. if you know distributional properties of one series you immediately know the distributional properties of the other. In fact if your data have time series which are perfectly colinear it usually means that one of the series was artificialy created, for example: profit equals income minus expenses.