No, you only get the best one
From the Druid's Wild Shape feature:
You retain the benefit of any features from your class,
race, or other source and can use them if the new
form is physically capable of doing so.
Which (as the Elk form also has a charge feature) is quite clearly applicable. Therefore in Elk form the centaur feature is still available. However, from the DMG p. 252 (provided in the errata, thanks NautArch):
Different game features can affect a target at the same time. But when two or more game features have the same name, only the effects of one of them—the most potent one—apply while the durations of the effects overlap. For example, if a target is ignited by a fire elemental’s Fire Form trait, the ongoing fire damage doesn’t increase if the burning target is subjected to that trait again. Game features include spells, class features, feats, racial traits, monster abilities, and magic items. See the related rule in the “Combining Magical Effects” section of chapter 10 in the Player’s Handbook.
The Charge features are included in this and so only the best one is applicable (if it is ambiguous which the best, you choose).
The kind of situation this rule prevents, in addition to the same spell and similar Auras etc., is that if an Elk gains 6 Druid levels and takes the Circle of the Moon (let's not worry how) and Wild Shapes into a Giant Elk it would otherwise have two instances of the same charge feature (admittedly with different DC's). If both of these were to apply the attack in question would deal and extra 4d6 and cause two Strength saving throws against being prone'd.
In the specific example there are two different features (Centaur and Elk charges), but with the same proper name (Charge) and so only one of them should apply.
A character ending up with two copies of the same feature is otherwise avoided by the games rules trough the special rules for multiclassing which would otherwise be the way to obtain those. For example if two of your classes would give you the Extra Attack feature, only the best one applies or of your would gain a feature called Unarmored Defense and you already have one, you simply don't get the new one (even if it uses a different AC calculation). (See PHB p. 162)