Wizkids Rules!

Doomsday
At the beginning of the game, choose a character die's name.
When fielded, KO target character die with the chosen name.

Moon Knight
At the beginning of the game, choose one of your Avengers character cards. For the remainder of the game, Moon Knight is considered to have the text of that card (including Global Abilities) instead of this text (replacing their name with his own, if applicable).

With the addition of abilities like Doomsday and Moon Knight, players are now allowed to make selections at the beginning of the game. If both players happen to bring cards with abilities like these, does each player make their choice in secret then reveal the choice? Should the choice follow player order with the First Player making their selection and the Second Player making their choice after? Is there another way this should be decided?

Very interesting niche question.

Technically, the beginning of the game commences with the determination of who gets to choose who goes first. So, the ‘First/Second player selects’ option is moot.
If both players bring cards with these timing abilities, each player looks at their opponent’s team (if necessary), makes their choice secretly, then the chosen cards for those abilities are simultaneously revealed.

We leave it to the TO or judge to determine the exact method. Suggestions: each player writes their choices down and reveals them simultaneously; each player secretly tells the TO/judge their choice, then the TO/judge announces the choices, or any other fair method.

Whatever the method, the intent is for neither player to gain an advantage through the selection and announcement of their respective choices.