If you collect all the water expelled while a laundry machine washes a load of clothes, all of the synthetic material will be in the water. If you filter that water with the appropriate filter, all of the synthetic material will be captured by the filter. Unless the clothes washed were very dirty, most of the material captured will be fibers from the clothes. The synthetic fibers will be mixed with organic fibers (cotton, wool) and it will not be simple to separate them. If only pure synthetic clothes are washed, all the captured material will be synthetic. Drying and weighing the clothes and then the captured material will tell you the proportion of material that is lost with a single washing.
Synthetic fibers from clothes can be a problem for septic systems because they never decompose and eventually clog up the system. Municipal sewage treatment facilities separate the solids from the effluent and most of the solid material is not discharged into waterways. So it might be that the bulk of the fibers never make it to the ocean. Microbeads are small enough that they might stay suspended and pass through filters, so some of them can reach the ocean. Microbeads are added to various products but generally not clothes.
Another fun exercise might be to dissolve some soap or skin products to see if you can separate microbeads by filtering the water. Some of these products will not dissolve in water so other solvents would be required.