@Walker notes below in their comment that in UK red is hot, black is neutral and copper is ground from pole. My comments below all assumed this was a US installation, with red and black opposite hot phases of 120V and copper neutral; OP indicated they measured about 110V between each of red and black and "ground", so this seems likely to be US usage.
Some clarification just in case:
The cables coming from the pole into the shed brings in two wires - one red, one black, and is wrapped in many small copper wires that collectively act as the ground (or common?)
The many small copper wires are almost certainly the common/neutral, NOT ground. They provide the electrical return path for the incoming "hot" red and black wires. Voltage measured red to neutral or black to neutral in the US should be 120V. The red and black are different "phases" of hot and the voltage measured between them should be 240V.
For "grounded outlet" safety, the "equipment ground" wire (e.g. the third prong in a 3 prong plug) is wired back to eventually connect to the ground bar in the main electrical panel. In that main/first electrical panel ONLY, the ground bar is bonded (electrically connected) to the neutral bar.
This is so if e.g. an appliance metal frame accidentally comes in contact with a "hot" wire, the current immediately flows through the equipment grounding wire back to the main ground bar and then neutral as a short circuit, and the breaker trips, shutting off the power and thus making it safe to touch. This normally does not involve the "earth ground" (e.g. grounding electrodes 8 feet into the ground or equivalent).
Should I ground the box to the common?
All electrical panels/boxes should have both the metal enclosure and the grounding bar (if present) connected to the equipment grounding conductor (not the neutral/common).
In the main (first) panel/enclosure ONLY you should "bond" (electrically connect) the ground bar to the neutral bar.
In that main panel/enclosure you should also bond an earth ground (known as Grounding Electrode Conductor, usually attached to e.g. grounding electrodes in the ground) to the grounding bar. The GEC may also need to be bonded to the metal enclosure where it enters that enclosure.
Adding a grounding bar to your box
It looks like your box may be Square D Homeline brand. They make add-on grounding bars like the one below:
I believe a bar like the one above would be mounted to the two pre-existing holes near the bottom right of your photo, one above the other, on the small "bumps". This size bar would give you six lugs to attach wires; I believe longer bars are available if you need more. You would attach your Grounding Electrode Conductor to one lug, and each circuit's Equipment Grounding Conductors to other lugs. The two screws supplied with the grounding bar would electrically bond it to the metal enclosure.
I believe that according to recent code you need something like a Kenny Clamp:
to bond your Grounding Electrode Conductor (which should be 6 ga copper or larger IIRC for 100A service) to the box as it enters from the outside. The Grounding Electrode Conductor needs to be unbroken (not spliced) from its connection to the first/main grounding electrode all the way to its termination at the ground bar. The Kenny Clamp allows an electrical connection at the box entrance without interrupting the GEC.
IIRC, the grounding electrodes need to be at least 8 feet long in the ground and at least 5/8" diameter, with at least 2 spaced 6 feet apart. They are easily available in home centers/hardware stores as pointed copper plated steel grounding rods. Don't use the short skinny 4 foot ones as those are not Code compliant.
Finally, to bond the neutral/common bar to the box and thus to ground, make sure the connection is a big enough size to handle the maximum expected current. I am not sure of Code on this, but suspect you will need larger than 12 ga; Square D should have an appropriately beefy bonding clamp/solution available.
Regarding the neutral bus bonding screw to the metal enclosure, I found this set (use one): Square D part number PKNBSCP
The empty hole among the lugs on the neutral bar is for the bonding screw. Such a hole will have a hole in the panel skin underneath, allowing for the self-tapping bonding screw to make a good connection, using substantial turning force on the screwdriver.
Found the panel model, I think:
This seems to be an older model, labeled "Square D - HOM612L100SCP Homeline Load Center with Cover, 100-Amp Fixed Main Lugs, 1-Phase, 6-Space, 12-Circuit, Indoor"
Wiring instructions are said to be on the label barely visible on the left inside wall of the enclosure, as ThreePhaseEel suggested.
The add-on grounding bar kit (up to 4 can be mounted using the 4 pairs of mounting holes at the corners of the back panel):