I thought the app was just for bt broadband customers but may be wrong
Although BT Sport is free to BT broadband customers, the Terms and Conditions present it as a pay TV service. Paragraph 3 states that whichever TV platform you choose [which would have to be Sky for a non-BT BB customer] you may also access BT Sport content via BT Sport Online (the BT Sport App or the BT Sport Website). Paragraph 46 reiterates this.
That seems to add up to a "Yes".
Yes you are correct, this was asked on facebook the other and the same answer came back as you have pointed out