At least in the social sciences you often have panel data that has large N and small T asymptotics, meaning that you observe each entity for a relatively short period of time. This is why applied work with panel data is often somewhat less concerned with the time series component of the data.
Nevertheless time-series elements are still important in the treatment of panel data. For instance, the degree of auto-correlation determines whether fixed effects or first differences is more efficient. In difference in differences proper treatment of the standard errors to account for autocorrelation is important for correct inference (see Bertrand et al., 2004). Dynamic panels using estimators for small N, large T asymptotics are also available, you often find such data in macroeconomics. There you may run into known time-series issues like panel non-stationarity.
An excellent treatment of these topics is provided in Wooldridge (2010) "Econometric Analysis of Cross Section and Panel Data".
I'm just thinking out loud here,
Suppose you have industry-county-year level data, your outcome is $Y_{ict}$, and you are interested in the effect of some variable $x_{ict}$.
In your strategy you would correctly think you can use:
(1) industry-county (panel) fixed effects to control for time invariant confounding factors across these panels as well as the average difference in time varying covariates across industry-county pairs
(2) year fixed effects to control for shocks that are common to all industries and counties in a given year
However what if there are shocks that are common across some counties in regions indexed by $r$, yet are both time varying and different across regions?
That is, perhaps the true data generating process is
$Y_{ict}=\underbrace{\theta_{ic}}_\text{panel fixed effect}+\underbrace{\theta_t}_\text{year fixed effect}+\underbrace{\theta_{rt}}_\text{regional shocks}+\underbrace{\beta}_\text{parameter of interest} X_{ict}+\underbrace{\epsilon_{ict}}_\text{idiosyncratic shock}$
But you estimate a model
$Y_{ict}=\theta_{ic}+\theta_t+\beta X_{ict}+\epsilon_{ict}$
which does not attempt to proxy for this regional shock, then,to to the degree that $Cov(\theta_{rt},X_{ict})\neq 0$, I believe your estimate $\hat{\beta}$ would in part reflect the variation in $\theta_{rt}$ that covaries with $X_{ict}$.
That is,
$plim \; \hat{\beta} =\underbrace{ \beta}_\text{true parameter} + \underbrace{\frac{Cov(X_{ict},\theta_{rt})}{Var(X_{ict}}}_\text{bias}$
to solve this I believe it is possible that you could
(1) Cluster your standard errors at the geographical level where you think there may be correlated disturbances
and
2) Find an instrument $Z_{ict}$ for $X_{ict}$ that is strongly correlated with $X_{ict}$ (relevant) that has an effect on the outcome only through its effect on $X_{ict}$ and not through $\theta_{rt}$ influencing $Z_{ict}$ or through $Z_{ict}$ influencing $Y_{ict}$ directly (excludibility).
Best Answer
In short, there is no such thing as Multivariate Time Series data. The only classic data types out there are: Cross Sections, Time Series, Pooled Cross Sections, and Panel data.
Panel data is multidimensional. Time series is one-dimensional. Time Series data is a type of panel data. Daily closing prices for last one year for 1 company is a Time Series dataset because the time variable alone uniquely identifies each observation. I will work with only 8 days worth of prices instead of a year, but the idea is the same.
Daily closing prices for last one year for 10 companies can be both a Panel dataset and a Time Series dataset, depending on whether it is in wide or long format.
If the data is organized so that Time still uniquely identifies each observation (i.e. wide format) then it is still a time series dataset, and you will have different columns for the daily closing price for each of the 10 companies. See example below:
If, on the other hand, it is organized so that Time alone no longer uniquely identifies each observation (i.e. long format) then it is a panel dataset. With 10 companies, you need to use Time and Company ID together to uniquely identify each observation. See example below:
In any statistical software it is straightforward to reshape a dataset from wide to long and from long to wide.
The term Multivariate Time Series is heard around, but it refers not to a type of dataset but to a type of analysis. To be more exact, it refers to the type of time series regression analysis in which there is more than one response variable. (Make sure to distinguish it from Multiple Time Series regression, which refers to a regression with one response variable and several predictor variables). One example of Multivariate Time Series Analysis is VAR (Vector AutoRegression).
More on VAR here: https://en.wikipedia.org/wiki/Vector_autoregression
I hope this helps.