In the form of the "product" module, I need to add the fields for translations of this name into other languages, in my case into German, English, and Ukrainian, below the name field.
Yeah, you could do it by creating several computed fields to be able to handle the name translations per lang. One field per lang and implement the compute and inverse methods of the field to retrieve and save the translation values directly into the ir_translation table
You can add a new field(s) and show translated values in it. Make the new field as a computed field, which computes the translated value from the saved translation.
While doing the compute you can pass the lang in context.