IIRC Oracle does support this and calls it a skip-scan. If the first column in the index is not very selective this can be very efficient, though if the first column is good in terms of selectivity then a skip-scan will be very inefficient and the DB might as well just scan the whole index. Given that a good choice of index usually has a fairly selective column as its first, and that unless storage space is very cramped you are likely better off just defining an extra index without the column that you want to not touch in some queries, this means that a skip-scan isn't helpful as often as one might think which is one of the reasons DB engine maintainers give for not spending time implementing and maintaining the feature.

There are some use cases where having the option to skip-scan is definitely a significant bonus, but they are rare enough that I understand why most engine maintainers have not implemented them.