From my understanding, the Brønsted-Lowry (BL) acid/base theory is in relation to the idea of conjugate acids and bases.
A conjugate base does not always occur when there is interaction between a Lewis acid and a Lewis base, but a conjugate base always occurs when there is a BL acid-base reaction. This is because BL acids-bases are defined by the exchange of protons (H+ ions).
When a BL acid donates a proton to a BL base, the result is a conjugate base (originally the acid) and a conjugate acid (originally the base). And, you must notice that this reaction is in equilibrium! Thus, the reverse reaction is possible. This means that the conjugate base will act like a BL base.
Now, it may be that the initial BL acid that was used was considered a very strong acid, so the conjugate base will be a very weak BL base. Thus, the equilibrium may be favoring the products (K>>>1). As a result, in a closed system, very little conjugate base will undergo the reverse reaction. Nevertheless, at least some should undergo the reverse reaction and it will then act as a BL base!