Let's say I have a male + female connector which are rated up to 250
volts and 5 amps. This should then be capable of handling 1250 watts
as far as I know.
Now, when using lower voltages such as 12v, in order to reach 1250
watts, it should be around 104 amps.
Connectors do not have a constant power rating. The voltage and current limits apply independently of each other. So your connector can handle a maximum of 5 A at any voltage from 0 to 250 volts and it can handle a maximum of 250 V at any current from 0 to 5 amps. The only way it can handle 1250 W is to run it at both 250 V and 5 A. The current rating doesn't increase when you run it at lower voltages (and the voltage rating doesn't increase when you run it at lower currents).
I'm assuming here that the connectors stay mated, i.e. plugged together. The ratings for making (plugging together) or breaking (pulling apart) a connector while the load is active can be smaller. To find out what that is you'll need to look at the data sheet for your specific connector, if it's even allowed at all.
Let's say you make a circuit which uses 12v@10A, which is about double
the current rating of the 220v@5A. ... Do I need to find a different
connector.
Yes. You need to use a connector rated for at least 10 A at least 12 V. You can't use a 5 A connector.
Also, for safety, if a connector has an expected use you shouldn't use it for something else even if that's within the connector rating. For example, an AC grid/mains connector shouldn't be used for something else because at some point somebody, possibly yourself, will plug it into the AC supply and cause injury or damage.