It's tapped like that because that's the way it was supposed to be done when that work was done. Back in those days (<2011) if you had a single 2-wire feed (romex) to and fro a switch box. The white wire would, according to more recent codes prior to NEC 2011, and possibly then too but not followed, have been taped a solid color other than green to notify you that it is indeed hot.
This is done to ensure the wire feeding power to the light is black as it would/should be in normal situations; it's safety and economy. The other way to deal with this is to pull a 3-wire feed to the switch. Today this is required not to rid the world of the marking style, as it is still permitted under certain exceptions today, but to cover certain types of switches which require it. As recent as NEC 2008 it was not mandatory unless the manufacturer of the device required it.
You should tape it up (the white wire), then replace all those taped joints with wire nuts appropriately sized for the taps.
There may be some challenges installing the meter you have selected in a code compliant permanent installation.
Regarding the connection of the meter's voltage leads, using #12 - #18 wire:
Tap the terminal screws on the receptacle? This one is a non-starter. You'll be violating the listing of the device and the bad connection is very likely to cause serious trouble.
Wire nut pigtails? The manufacturer's documentation for wire nuts will explicitly specify the wire combinations for which it is approved. There are large wire nuts that work with two #6's and a #12. I don't think you'll find one rated for two #6's and a #18. However be advised that making a good splice with a wire nut does take some skill.
Split bolt? As mentioned in the comments a tap and run split bolt connection is very secure. Again you'll have to use a split bolt rated for #6 run and #12 tap, and I doubt you'd find one rated for #6 run - #18 tap. Skinning the #6 and insulating the splice again takes some skill.
Distribution block? A distribution block or terminal block rated to handle the amperage of the circuit and with terminals rated to accept #6 and #12 conductors will work well and is easy to use. It will take up some additional space inside the enclosure. You can even find block rated to handle 65 amps with terminals that handle #6 and #18 wire. Still, I'd use #12 for other reasons (read on...)
A similar solution mentioned in the comments is a multi-tap connector like the Polaris Insul-tap. It is like a distribution block but fully insulated so it need not be mounted.
- Subpanel? This is a creative solution, and an exepensive one, it avoids some code issues but may raise others, I'd say that's a whole separate question.
Another creative solution - technically taped solder joints are still permitted in the NEC. Nobody seriously considers doing this for building wiring any more but I have to say when you come across these in old buildings, they have stood the test of time. I wouldn't suggest reviving this old method for power wiring.
Another solution would be a crimp on C-tap for #6 run / #12 tap. This makes an excellent connection, and although it has to be taped like a split bolt, it's easier because it's less lumpy. Although the C-taps are not expensive, the crimp tool that you use with them is quite expensive.
If you keep the meter voltage leads short - under 18" - and use #12 wire, I believe you can use NEC 210.20(4) Exception #1 (C) to tap the 50A circuit without adding an overcurrent protection device (breaker or fuse) to the meter leads. The meter would be an individual, non-receptacle outlet.
Exception No. 1: Tap conductors shall have an ampacity
suffıcient for the load served. In addition, they shall have an
ampacity of not less than 15 for circuits rated less than 40
amperes and not less than 20 for circuits rated at 40 or 50
amperes and only where these tap conductors supply any of
the following loads:
(a) Individual lampholders or luminaires with taps extending
not longer than 450 mm (18 in.) beyond any portion
of the lampholder or luminaire.
(b) A luminaire having tap conductors as provided in
410.117.
(c) Individual outlets, other than receptacle outlets,
with taps not over 450 mm (18 in.) long.
(d) Infrared lamp industrial heating appliances.
(e) Nonheating leads of deicing and snow-melting
cables and mats.
Another exception
Exception No. 2: Fixture wires and flexible cords shall be
permitted to be smaller than 14 AWG as permitted by
240.5.
might allow you to install a second 14-50 receptacle, and make up a flexible cord and plug for the meter voltage leads. I don't like this idea, but it might be code compliant.
Taking a step back, I might be a bit concerned that the meter in your link is listed and suitable for permanent installation.
Best Answer
If you are discontent with the quality of wire nuts, you can go to a proper splice block. These are my go-to for anything larger than 6 AWG.
The wires are inserted in the side, then they are tightened down from the top holes with hex keys to a specified torque. The holes where the hex keys go in are then capped. If the wires are near the largest size the block permits, there is nothing exposed at all. The quality of the connection is exceptional.