No. The “chasing arrows” symbol is simply a communication tool used by manufacturers. Sometimes it means a manufacturer used recycled material in the product. Other times it shares information about how the product is made. For instance, the chasing arrows symbol with a number indicates the plastic resin code—meaning the base type of plastic being used. But plastics are made in different ways, and use various additives, which give them their shape, flexibility, strength, etc. So, not all plastics with a particular number (ex: #1 PET) are the same.
Follow the Big 3 guidelines only when deciding whether something should be placed in your curbside bin. You can use the chasing arrows symbol as a guide when buying materials, as you look for items that include recycled content in the product.