It depends on your modelling / views. For a given time series with a timespan $T$, you can consider that you observe $T$ realizations of given random variable $X$, or your can consider that you observe one realization of a stochastic process, that is one path among many others. If you consider a independent identically random process, these are the same.

It is not clear, $Y_1,Y_2,\ldots,Y_n$ could represent the $n$ variables of your random process and thus a single time series, or $n$ time series each represented by a single random variable $Y_i$, and therefore your time series data is a matrix $n \times T$, i.e. a time series for each $Y_i$ with $T$ realizations.

As long as your consistent, it is up to you to choose your modelling.

It depends on your model / views. For a given time series with a timespan $T$, you can consider that you observe $T$ realizations of a given random variable $X$, or you can consider that you observe one realization of a stochastic process that is one path among many others. If you consider an independent identically distributed random process, these are the same.

It is not clear whether $Y_1,Y_2,\ldots,Y_n$ represents the $n$ variables of your random process and thus a single time series, or $n$ time series each represented by a single random variable $Y_i$, and therefore your time series data is a matrix $n \times T$, i.e. a time series for each $Y_i$ with $T$ realizations.

As long as you are consistent, it is up to you to choose your model.