WM doesn't accept other stores' coupons. In their coupon policy they talk about accepting competitor coupons that are issued at the register by Catalina, those are still manufacturer coupons but they usually have other stores' logos and may say "redeemable at Walgreens".
They will also accept "competitor coupons" like Wags in-ad coupons that state the product and it's price. But that is essentially price matching - they'll match the price stated on the coupon.
WM does NOT accept competitor "dollars and cents off" coupons, like Target web coupons and others. HTH.