There is still a faint climate signal in there. Moving from March to September shifts the embeddings by an amount that roughly matches climatic intuition: tropical and humid zones move a lot, dry deserts barely budge at all. The authors are careful to call this a weak signal that is probably driven mainly by geography rather than by any real grasp of climate.
Evidence
correlational
Key metric
seasonal shift magnitude Af 7.59, Cwa 7.41, Am 7.17 against BWh 1.67 and BWk 1.89; mean cosine similarity between all pairs of shift vectors 0.32
Caveat
The authors describe the signal as weak and attribute it mainly to geography rather than to a representation of climate.