1. We got close to penguins.

2. We got near penguins.

(Does #1 mean #2? close to = near?)

Yes. Those sentences have the same meaning. "Close to" means "near."

You should probably add "the" before penguins. It sounds much better.

Yes, in this context, "close to" and "near" can be used interchangeably. Both phrases imply proximity or being in close proximity to something or someone. Thus, "We got close to penguins" and "We got near penguins" convey a similar meaning that the person or group of people approached or came in close proximity to the penguins.