Does a high-dimensionality of a time series refer to its length, number of variables or both?

I'd like to cluster some time series that describe a flow of a variable (say, temperature) throughout a day. Measurements are made every 5 minutes so each time series has 288 values.

Are we talking of high-dimensionality in this case? Will the euclidean distance perform well if I'd like to cluster my data with k-means? Should I find a way to reduce the dimensionality?

More specifically, does a high-dimensionality of a time series refer to its length, number of variables or both?

There are some other topics that touch this topic but there is no direct answer: yes - no. There is also a quite similar question on the Cross Validated but unfortunately it hasn't been answered: Curse of dimensionality for time series?

for instance, you may have process (SABR volatility): $$dF_t=\sigma_t F^\beta_t\, dW_t,$$ $$d\sigma_t=\alpha\sigma_t\, dZ_t,$$ where the only observable is $$F_t$$. However, it is at least a two dimensional process where you have at least two sources of noise $$W_t,Z_t$$. The volatility (of volatility) $$\sigma_t$$ is not observable. Similarly, state space models will have more dimensions that observable series.
One (more) observation, however, corresponds to having one (more) data point in the $$p$$-dimensional plane, where $$p$$ is the number of features.