Can I offer the buyer a choice between several logos?

Short answer: Yes. In the product configuration, turn on “Allow the buyer to choose between several logos”.

Steps

On the “Shops” page, open a draft of the shop — a live shop is never modified directly (orange “Modify the shop (create a draft)” button).
In a collection, click the product to open its configuration.
Tick the “Allow the buyer to choose between several logos” checkbox.
Go back up to the placement concerned: you can now click several logo thumbnails instead of only one. Select every logo the buyer will be able to choose — a small number appears on each one to show the order they are offered in.
Save.

The buyer will then be offered the choice when ordering.

As long as only one logo is selected on the placement, the label “No choice of logo!” is displayed: the option is on, but there is nothing to choose. As soon as there are two or more, the label becomes “Choice between N logos”.

Do not confuse this with using different logos on different products — there it is you who decides, not the buyer. See Can I use different logos on different products.

You'll know it worked when: The product page in the shop offers the choice between the logos you configured.