Latin vēr and Greek éar are inherited from the same Indo-European word for spring, and the Romance names descend from the Latin. The English word spring is unrelated and much later — it originally described what the plants do, not the season, and only became the season's name in the sixteenth century, displacing Lent.
Persian bahār is the most productive element here, and its cultural weight comes from Nowruz, the new year at the spring equinox, which is at least two and a half thousand years old and is still the central festival across Iran, Afghanistan, Central Asia and the Kurdish regions. A Persian spring-name is attached to a specific and living celebration in a way that the European equivalents mostly are not.
Japanese haru is similarly productive, and carries the association with cherry blossom that shapes so much of Japanese seasonal naming.
The traditions with two seasons rather than four name differently again: across much of South Asia and the Sahel the meaningful division is wet and dry, and what a European would call spring-naming is closer to the rain page.