Tiny scorch in plug – could be from fuse going, or from the intermittent overloading it’s probably had at times when both appliances are running and drawing a lot of current.
A 13A fuse (any wire fuse) doesn’t have a sharp cut-off. Most will probably never blow at 14A; at 20A it’ll probably take hours; at 30A probably a few seconds to minutes; at 200A under a second. Been a while since I looked at the numbers, so these may be a bit off, but you get the idea – the fuse won’t normally blow if you overload it a bit, for a short time.
As mentioned above, plugs / wiring may get quite warm at times with your setup. Try and find the rating plates on the back of the appliances. These will tell you the total power (watts, W or kilowatts, kW) and current (amps, A) required by each.
If the total for the two appliances is more than 13A, or 3000W (3kW), then running them both from a single extension is not a great idea.
How much hassle would it be to run a second extension?