How does Underrealm Lich interact with Sidisi when drawing multiple cards?
You get 3 Zombie tokens.

Each card draw is processed separately from the others:


  120.6b If an effect replaces a draw within a sequence of card draws, the replacement effect is completed before resuming the sequence.


As you already noted, you can only draw one card at a time:


  120.2. Cards may only be drawn one at a time. If a player is instructed to draw multiple cards, that player performs that many individual card draws.


The draw replacement effect ultimately causes you to move cards from your library to your graveyard ("move" for brevity). Since you can only draw one card at a time, and the replacement effects works on each draw individually in sequence, you also move 2 cards at a time, and Sidisi triggers on each of those moves.

Since the card draw is the event that, together with Lich's replacement effect, ultimately triggers Sidisi's ability, consider an analogous situation: Underworld Dreams triggers on each card draw. If you were to draw 3 cards, you would get 3 separate damage events, as the ruling on that card's gatherer page confirms. Therefore, any ability that is triggered by a replacement effect that replaced a card draw would react the same way.