The Common Thank-you Email (defined in Seller Admin > Edit Preferences) gets sent to the buyer for every order regardless of which item(s) they purchased, and it is unrelated to any specific item's Product Thank-you Email, which gets sent separately only when that item is purchased. The Common and Product emails are separate messages, each with their own settings that don't interact. The Common email always supports template tags, but it cannot use any item-specific tags because it is not associated with any particular product and only gets sent once per order even for multi-item orders.
Also note that using "Send free download link" will only send the selected item's Product Thank-you Email, if one is enabled for that item; it will not also send a Common email, which only gets sent for actual orders. If you enter a custom Email Message whens ending a free link, that would be sent instead of whatever Product Thank-you Email (if any) you have defined in that item's settings.
This help page explains more about thank-you emails and how to customize or disable them: