If you have $n$ time series of rainfall measurements every hour ($x$=time, $y$=amount of rain), and compute the distance matrix between each pair of time series based on Dynamic Time Warping, and then plot the clusters, what do the X and Y axes on the cluster plot represent?
They remain the same, $x$=time, $y$=amount of rain - dynamic time warping (DTW) shifts the times so the curves of the amount of rain align their peaks and valleys.
A different example might be easier to understand. In the paper by Berndt and Clifford (see below) the population of the Snowshoe Hare is plotted and subjected to DTW. You can see as the population of hares increased so did the ability to hunt Lynx. As the population of Lynx increased the population of hares declined. That affected the hunting of Lynx, which again allowed the hare population to increase. This creates a double-top peak in the data, using DTW these can be aligned.
In the k-medoids or k-means plot, is X still time, and Y still the amount of rain?
Having aligned the time/population (similar to time/rainfall) the locations on a map can be plotted with a k-medoids or k-means plot.
K-medoids (also: Partitioning Around Medoids, PAM) uses the medoid instead of the mean, and this way minimizes the sum of distances for arbitrary distance functions. K-means clustering aims to partition n observations into k clusters in which each observation belongs to the cluster with the nearest mean, serving as a prototype of the cluster.
The time variable would be plotted on the map for a period of time, for example a month, season, or year; so while the time is not 'lost' it's also not $x$ or $y$. Instead a color on the map could represent the amount at a location for the entire duration of the period of time.
See how when k-medoids or k-means is applied "time" does not sit on $x$ or $y$, in this example location is on the $x$ and $y$:
"Alignment of curves by dynamic time warping", by Theo Gasser and Kongming Wang.
"Using Dynamic Time Warping to Find Patterns in Time Series", by Donald J. Berndt and James Clifford.
"Clustering analysis applied to NDVI/NOAA multitemporal images to improve the monitoring process of sugarcane crops", by Luciana Alvim Santos Romani, Renata Ribeiro do Valle Gonçalves, Bruno Ferraz Amaral and Agma Traina.