China had no suzerain over Japan,Japan paid tribute between 600-838bc but that's for trade access to Chinese market,China had no administrative control on Japanese empress.
Both China & Japan wasn't unified back in Sui era, the big empire in china was in a contest against Goguryeo and there's Bekjae ,Shilla in north east asia.
Before Japan became more centralized under the central ruler ,various clans acted with a certain degree of independence,some clans had send tribute to china for trade access and got inferred titles to boost their standing, but the Japanese empire had no need for Chinese decree and the Japanese emperess made it clear from the letter send by Ono no Imoko, a Japanese politician and diplomat in the late 6th and early 7th century, during the Asuka period,appointed by Empress Suiko as an official envoy to the Sui court in 607 (Imperial embassies to China),delivered the famous letter from Japan's Prince Shōtoku which began "The Son of Heaven where the sun rises [Japan], to the Son of Heaven where the sun sets [China], may good health be with you."
A vassal living under Chinese suzerain would never write like that. Japan being a vassal of China is absolutely wrong ,Japan along wit Baekjae fought even against Tang-Shilla .
China -Japan had a Master -Student relationship , but saying Japan was a vassal is historically wrong.