Questions & Answers about Adiós; hasta mañana.
It is definitely used, but in Spain it can sometimes sound a little more definite or more serious than some other everyday goodbyes.
Among friends, people often say things like hasta luego, nos vemos, or chao. Adiós is still perfectly normal, but depending on tone and context, it can feel slightly more final than bye in English.
So in Adiós; hasta mañana, the second part softens that feeling and makes it clear that you expect to see the person again soon.
Because hasta mañana is a fixed farewell expression.
Spanish often leaves out words that are understood from context. Literally, hasta means until, and mañana means tomorrow, so the phrase is basically until tomorrow. In natural English, that usually corresponds to see you tomorrow rather than a word-for-word translation.
So even though there is no verb written, the idea is complete and idiomatic.