Actually, since there are several types of monoidal functors (lax, colax, and strong) there are several types of “adjunctions between monoidal categories which respect the monoidal structure.” Namely, we could have:

An adjunction in the 2-category$MonCat$ of monoidal categories and strong monoidal functors. In this case both the left and right adjoint are strong. We call this a strong monoidal adjunction.

An adjunction in the 2-category $MonCat_\ell$ of monoidal categories and lax monoidal functors. In this case the right adjoint is lax, while the left adjoint is necessarily strong (by doctrinal adjunction; see here). This version, which is one of the most frequently occurring, is often called simply a monoidal adjunction.

The dual: an adjunction in the 2-category $MonCat_c$ of monoidal categories and colax monoidal functors, in which case the left adjoint is colax and the right adjoint is strong. One might call this an opmonoidal adjunction.

A mixed situation, in which the left adjoint is colax, the right adjoint is lax, and the lax and colax structure maps are mates under the adjunction. This is a conjunction in the double category of monoidal categories and lax and colax monoidal functors, so we may call it a monoidal conjunction or a lax/colax monoidal adjunction. By doctrinal adjunction, given any adjunction between monoidal categories, if the right adjoint is lax monoidal, then the left adjoint automatically acquires a colax monoidal structure making the adjunction into a monoidal conjunction, and dually.