How much 12V DC current can I safely draw from an AC 230V 13A socket and why?
Power is current times voltage (P = IE). You don't mention if you're converting from one voltage to another. Are you using a step-down converter? Are you merely charging the battery with the solar system and want to know how to achieve equivalent power from a charged battery bank?

230V AC 13A is 2990 watts. 2990 watts at 12V DC would be ~249 amperes. This means your battery(ies) (and all the connectors and cabling) would have to be capable of safely delivering 250A in order to have roughly equivalent power to your UK mains example.

Because you said "13A at 12V can barely reach 150W" it seems like you're already aware of the relationship between voltage, current, and power. It also sounds like you're looking for a better socket to use for your system, and chose the UK mains style because it won't likely be confused with the real thing.

So here's what you are maybe missing, related to my question from 8 years ago about fuses. Voltage is what "motivates" electrons to go through a particular thing, whether it's a fuse, a wire, or a connector. The current is "how many" of them. The larger the current, the greater the friction, and thus heat. A connector or wire rated for 13A is not going to handle more power unless the voltage is also higher. In the case of a connector, the voltage rating will be mostly applicable to the distance between conductors (to avoid arcing) while the current rating will be applicable to the robustness of the conductors.

Put another way, 13A can have wildly different power values based on voltage, but it's always going to be the same quantity of electric current flowing. If the voltage is 12V, you're right, it isn't a lot of power, but it still requires thick wire and connectors to safely handle that current without having an unsafe temperature rise.

You will likely want to use something capable of much higher current. Automotive applications that use 12V systems often have fuses and wire rated for 50 or more amps. But be absolutely sure to look at the specifications for your battery (or battery bank). There's no point in installing 60A-capable wiring and connectors if your source can only supply 40A. Also be sure to install fuses to limit current to less than the weakest link in the system.