We have had our Pygmy palms for about 2 years and transplanted them in another area where we had a Bougainvillea plant there for 5-7 years. We dug the Bougainvillea plant out and planted our Pygmy palms. We notice lately a bigger fan type of palm growing in front of it. Is that normal to have another palm attach to our existing palms?
No, there shouldn't be another palm growing there unless it is an offshoot of your own plant, which can be removed. It's possible a different variety grew from a seed.
https://www.gardeningknowhow.com/houseplants/pygmy-palm/grow-pygmy-palm-trees.htm