Should we stir the brew?

In beer making, it's not uncommon to stir the yeast, "rouse the yeast" during the initial few days of fermentation - but only if you did not use enough yeast, or some other problem happened (like your temperature went very low).

This moves the yeast off the bottom of the vessel (where it "flocculated" to), and re-mixes it with your beverage. It helps kick-start a sluggish fermentation.

All that said, If you pitched enough yeast, I don't think you need to do it.

One of the problems that can arise, is that it's introducing oxygen back into the young wine (which the yeast removes). In the initial stages this is OK, but in the longer term, allows for the growth of spoilage micro-organisms, like acetobacter. Plus there's a chance the simple action of opening the vessel will allow the ingress of further micro-organisms.

If you are using grapes, you can perform a maceration, where you will stir the solids in your must (or punch down the cap). But this is before fermentation.

After the first racking, where you get rid of the gross lees. You end up with fine lees at the bottom of your container. Those may be stirred before the following racking (to keep them longer), as some think it adds to the body and taste of the wine. But this is not mandatory.

Another moment people will stir the wine, is before bottling as a way to degass. Again, this is not mandatory, but some people will do it as it helps getting rid of the remaining bubbles. However, the longer you store your wine before bottling, the less you will need to degass it.

There should be no need to shake or stir after fermentation has started.