Does it change anything if the seller's business is already structured with a holding company on top?

Yes — it changes what you actually need to decide you're buying. If the seller's operating business sits inside a subsidiary owned by a holding company, you have real choices: buy the shares of the holding company itself, buy the shares of the operating subsidiary directly, or do an asset purchase from the operating subsidiary. Each option lands you in a different place.

Buying the holding company's shares means you inherit whatever else sits inside that holding company — other subsidiaries, investments, or liabilities unrelated to the business you actually want — unless those are carved out first. Buying the operating subsidiary's shares directly is usually cleaner, since you're not taking on the holding company's own history, but it requires the holding company, as the current shareholder, to properly authorize and complete that transfer. An asset purchase from the subsidiary avoids inheriting either entity's corporate history but comes with the usual asset-purchase considerations around contracts and consents.

Before assuming you're just "buying the business," it's worth having a business lawyer map out the seller's actual corporate structure and confirm exactly which entity, or which assets, you'd end up owning under each option.