New seat, battery and protective foam squares. But the foam square glued to the seat pan (that when the seat is down, rests on top of the battery) has this strip of brownish, oily something that runs the width of the battery. The oily strip aligns to where I'm pointing to in the second photo. Any ideas what this oily stuff is or where it's coming from? Is it battery acid? Fumes from battery acid? Is there danger it will eat into my beautiful new seat pan? The battery itself is not oily like this.

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Is the vent tube blocked or pinched?

Negative. Runs down and out past the swing arm.

Was the battery cleaned before being installed? Maybe it was pre existing and the foam just picked it up. Does the stain have an odor?

Battery was new when installed and clean. The stain used to have a faint gasoline-y smell, but not anymore.

Maybe oil cap was set on battery during fill and foam absorbed some oil left behind?

If you're concerned it's acid...  mix a little baking soda in a little water and apply it to the foam...  if it bubbles up it's acid.  But it really looks like an oil stain to me.

Baking soda trick is a great idea. Tried it. No bubbles, so I guess it is oil. From where, I have no clue.

My guess is the breather strip that runs across the top of the battery is made up of components or chemicals that are not compatible with the foam material. I’ve seen that happen with other materials in my past. You could maybe test it by putting layer of cellophane or Saran Wrap over the battery and pinch it in place when you close the seat. Leave it for a duration long enough to see if it stops the effect.