Xjyuk

One of our characters created a centaur Eagle totem barbarian (level 3) - which I thought was a good idea. But to duck and weave in tight spaces, she'd need high AC, so she came up with a character wielding a lance in one hand and a shield in another. Whilst wearing scale-mail, she'd have a total AC of 18. She was using the Guildmasters' Guide to Ravnica rules for the centaur race.

Problem is, could a centaur be counted as "mounted" to use a lance in one hand and a shield in another?

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.

First clarification:

Asker: Are Centaurs considered mounted for the purposes of wielding a lance, for example?
Crawford: Nope.

Second clarification:

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.

Technically speaking, no.

Mounted is a specific situation, and you don't count as mounted unless you are actually mounted or have a special ability that says you count as mounted.

A DM might decide to allow this.

However, if they do, the lance becomes by far the best weapon for a centaur. It has weaknesses (in the disadvantage for fighting within 5 feet), but trading the shield for a sword could easily overcome that problem, and you automatically avoid the biggest problem a mounted combatant faces, which is that his mount is an animal and subject to all the problems that come from mounted combat, such as the risks to your mount's safety and considerations of physical abilities (like "Can your horse even fit in the dungeon?" and "Your horse can't climb up a thirty-foot sheer cliff, come on man.")

In general, I would tend to shy away from rulings that make one specific option always the best choice for a given race/class/etc, especially when it involves removing penalties that are normally present for making that choice.