Name of accessory to hold rope together

They are joined...with wood glue. No, seriously...it actually is "stronger than wood". Bandsawn lamination is about as close as you'll get for a "specific name", though it could also be done with a router and templates. Or a CNC router, for that matter, these days.

The efficient approach will result in this one and a dark one with a light section, by sawing in stacks, then shuffling. This also has veneer strips along the glueline.

IMHO that's much too small in diameter for comfort. Possibly also for safety in that material. The "safe working load" listed is almost certainly not one involving using the rope for supporting people or things above people, where it's typical to derate (or over-size) by a factor of 7 or more.

IMPE a 1 inch/25mm to 1.5 inch/37mm diameter rope is more typical for a "knotted climbing rope" in manila. That may be sold as 3 - 4.5 inch (circumference) rope depending how it's listed for sale.

I'm dubious of sisal for the job as it's somewhat more prickly (the major use I'm familiar with it from is as haybale twine - and that's hard on the hands.)

Plastic or plastic blend ropes will hold up to weather better than natural fibers, but can be damaged by sun over time.