The weblio definitions do emphasize the specific seven colors:
赤・橙(だいだい)・黄・緑・青・藍(あい)・菫(すみれ)（紫）の7種類の色. 太陽光線をスペクトルで分けたときに見られる色。
You know, ROY G. BIV (I’m impressed at how directly that maps across the languages)
In context though, it’s like saying “rainbow-colored” in English. If you talk about something that has those seven colors… then it’s rainbow-colored.
“All the colors of the rainbow” could also be a good, less clinical-sounding English dictionary definition, I think.