I would like to add to this question from our perspective as a PCB assembly company. So this is regarding the actual manufacturing of your product.
To assemble the bare PCB, all PCBA companies will put your individual PCB into a panel of multiple PCBS. On one hand, this is to reduce the handling times (like buying a carton of 12 eggs instead of individual eggs), on the other hand, this is to make it possible to run the PCB on automated conveyors during assembly. Generally speaking, most companies will want to have a 10mm frame around the panel without components to make sure the PCB can be transported on conveyors and to have sufficient surface area to clamp the panel in the various assembly processes (paste printing, pick-and-place).
Having a round design limits the PCB assembler in how they can connect the individual board to this panel, and it ultimately limits the ways in which the depanelization can happen later. We like to panelize PCBs with V-Cuts instead of tabs (also called catwalks). This is because the panel connection is quite stable during assembly and can quickly be separated with a machine that runs a blade over it to cut the boards out of the panel. If you have a round PCB, there are limited options to connect the PCB with a v-cut. If you add perforated tabs, those might be in the way during your assembly unless the depanelization was done with a very precise milling process.
If you would like to have a round shape overall, I found it best to straighten out the edges of the PCB. You lose a bit of surface for routing, but you ultimately enable your PCB assembly house to chose whatever panelization method best fits your quantity.
Here is an example image I just drew up:
A: Round PCB
B: Round PCB with straightened edges
C: Example of panel connection with straightened edges
A final consideration is efficient material usage of PCB base material. While this is not such an important consideration with small volumes or double sided standard PCBs, wasting PCB base material can have a punishing effect on your final product price for very high layer counts or special PCBs such as metal core PCB or rigid-flex PCBs where the cost of the base materials are a significant factor of your PCB price.
EDIT - Updated PCB panel based on Spehro Pefhany's comment: