What is the ruling on washing a limb with the intention that it is a Sunnah, then it becomes clear that it is an obligatory act, and is the ablution valid in such a case?

The washing performed by the person mentioned is valid and is not invalidated by his washing a limb with the intention of it being supererogatory (sunnah), even though it is in fact obligatory. This is based on the principle that purification is among the acts of worship for which the intention of a supererogatory act suffices for the intention of an obligatory act. As stated in Al-Majmou' by Al-Nawawi, acts of worship are of three types. Among them are those that require the intention of performing the act itself, but not necessarily the intention of it being obligatory or specific, such as Hajj, Umrah, and purification. Thus, if one intends a supererogatory act for them, it counts for the obligatory one.