If Ivern has Svelsongur, does the Svelsongur get to pick a tag as it's played?

No, the Svellsongur does not get to pick a tag when it is attached to Ivern, Friend to All.

Here is the breakdown of why:

1. Copying Text vs. Triggering Abilities: When you attach Svellsongur to Ivern, Svellsongur copies Ivern's printed text. This means Svellsongur gains the text: "As you play me, choose Bird, Cat, Dog, or Poro. I gain that tag."
2. "As you play me": This specific phrasing refers to the action of playing the card from your hand onto the board. Because Svellsongur is already on the board (as a piece of gear) when it is attached to Ivern, the "As you play me" condition is never met.
3. Timing: Even if the ability were a triggered ability (e.g., "When I enter play"), it would not trigger because Svellsongur is already in play when it is attached.

In short, the "As you play me" ability on Ivern only functions when you are playing the Ivern card itself from your hand. Since Svellsongur is not being "played" as a unit when it is attached, that ability does not activate.