Re the boring bulb test bit.
Some wires can "appear" to be live when tested by a meter, but as the meter does not put any load on the wire, it can give a false reading.
The wire might not be comming directly from it's appropriate fuse, but instead, be going via another piece of equipment.
By testing it with a bulb, earthed to a clean bit of bare metal on the body, you are putting it under load, and the bulb will only glow brightly if the wire is a genuine live feed.
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Re wire size
OK - firstly if you like playing rave or reggae, loud, on a powerful system, then no the wire is not up to it.
But for "normal" use it is fine.
This is not really a safety issue by the way, it is more that during very loud bass passages in the music, such as a reggae beat or rave drum and bass sounds, the wire cannot pass enough current to produce the volume, and the bass gets distorted. (btw, this in turn distorts the other ranges due to a thing called "modulation", but that is even more boring ( tho I can if you want

)
In my opinion, if you are
not using any external amps, then the supplied wiring is fine.
Look at the thickness of the main + wire comming out of the radio and consider that as OK ( the radio manufacturer did ! ).
More detail =
The power ( measured in Amps ) that must go down the cable roughly equals the TOTAL power output in Watts, divided by the Volts ( 12 ) then mutiplied by 1.5.
Therfore a 100 watt stereo will use 12.5 Amps MAXIMUM.
The problem is that nowadays manufacturers tell lies about the power output of their stereos to sell them.
If it does not have a separate power amp it is
very unlikely to be more than a total of 80 Watts ( 4 outputs at 20 Watts to each speaker )
despite what may be printed on the box or leaflet.
Plus there are many different ways to measure power, RMS is the scientific way but gives the lowest figure so makers tend to use meaningless figures such as "Peak Power Output" or even "Power Peak Input".
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Re what wire to use.
Common sense plays a big part here.
If you feel the need to upgrade the wiring, then you might as well get some nice thick stuff as it is not much dearer than everage stuff.
Mains cable is fine, especially as it is useually double insulated so is less prone to chaffing through against rough metal edges ( though you should avoid this of course anyway, with rubber grommets etc ).
Plus you can twist the 2 or 3 wires in the mains cable together to double or triple the power handling.
Always use earth wires, from the radio (or amp), that are as short as possible and go to a nice bright good connection to a bare bit of metal bodywork. This will help eliminate unwanted noises. Don't forget this wire needs to be as thick as the power wire !!
If adding wires to the battery
ALWAYS stick a suitable high quality "inline" fuse, right close to where the wire is connected to the + positive terminal on the battery.
All this is for wiring a "normal" stereo.
If you are wiring a huge "max power" type 100000 Watt system, then you have more boring stuff to learn ( I can go on if you want

- or have I gone on long enough, probably)