The legal name of a business is the name of the person or entity that owns it. This is the legal trading entity.
If the business is a sole trader then the legal name is the last name with or without initials or forenames of the owner.
For unincorporated partnership, the legal name is the last names with or without initials or forenames of all of the partners.
For limited liability companies, partnerships and corporations, the business’ legal name is the one that was registered with Companies House including Ltd, LLP, PLC etc.
A trade or business name is the name a business uses for advertising and sales purposes that is different from the legal name described above.
Although a trade name may sometimes also be a trademark, a trade name is not, in itself, a form of intellectual property.
It is a requirement that businesses do not hide behind a trading name so whenever one is used there is a requirement to disclose the legal name and an address at which documents can be served on the business (for registered entities there are other disclosure requirements as well).
If you are unable to easily find out the name and address of a trader I would strongly recommend not doing business with them.