How to properly center the rear axle?

Visual is the right way. You do it by finding confluence between the tire appearing perfectly behind the seat tube, the gap between the chainstays and the tire being equal, and the gap between the seatstays and the tire being equal. Every once in a while you see a bike where those things can't all be true at once even with a correctly dished wheel, and that can indicate alignment problems, but it's not common. In a more practical sense, if your brake adjustment was dialed before taking the wheel off, you're really just putting it back where the pads want it to be. Grease everything and use either washers or track nuts to eliminate squirm.

Horizontal dropout bikes sometimes use their horizontal-dropout-ness to allow looser tolerances with chainstay cut length and potentially other manufacturing steps. If you were to attempt to precisely locate the axle at the same depth in the slots side to side, you'll see this sooner or later. Basic bike boom bikes did this quite a bit.

I've occasionally checked centering with a vernier caliper when I have cause to make things especially perfect, but it's not usually necessary.

In this case, you have a solid axle presumably with flanged nuts on the outside of the dropouts.

You're good as long as there's:

• 100% thread engagement on both nuts when everything is tightened
• No frame or brake rub

This means that the visible stick-out of thread on each side must be longer than the sum of the dropout thickness plus the nut's thickness. You might have a claw adapter or a derailleur protector or a rear axle kickstand to include on one side.

This does NOT apply for QR axles which are shorter and cannot exceed the dropout thickness plus any small gap under the acorn nut. That's a separate question though.