Fuses (and circuit breakers) are rated such that they blow or trip
before any part of the circuit gets to a dangerous current. You rate the
fuse or breaker based on the lowest current rating of all the devices
and wires in the circuit .
In your case, the manufacturer of the connection unit determined
(somehow) that the "weakest link" in that circuit can only draw 5A
before it exceeds its limits. Putting in a 13A fuse would thus allow
that weakest link to draw almost 3 times what it may be designed to,
which leads to increased heat and then fires.
If you double the size of the fuse, there is a small risk that the fuse will not blow if the transformer and circuit are not large enough to pull the amount of current the fuse is rated for (even if there is a short). This is similar to the scenario where an amp is pushed so hard that it clips (essentially runs out of gas because the transformer/power supply is not large enough to supply the expected current).
