A creature is only considered mounted if it is riding something
The rules for mounted combat (Player's Handbook, p. 198) say:
A willing creature that is at least one size larger than you and that has an appropriate anatomy can serve as a mount, using the following rules.
So, the centaur is not considered mounted if it isn't riding something. His racial features don't say it counts as "mounted" as well, so it can't wield a lance with a shield when all by himself.
However, the Centaur can still ride other creatures, as nothing in its features prevent it do so. Then, it would count as a mounted creature.
The official D&D lead rules designer, Jeremy Crawford, has clarified this twice, although his tweets no longer serve as official rulings.
Asker: Are Centaurs considered mounted for the purposes of wielding a lance, for example?
Asker: A fellow DM friend and I were discussing the if the centaur counts as being mounted at all times. Specifically for the use of one handing a lance. Could you clarify?
Crawford: If you’re not on a mount, you’re not mounted, unless a rule says otherwise.