Radiators for the automatic version of my car have a couple of extra coolant outlets at the bottom for cooling the auto transmission. I was sent the wrong radiator and I don't need these in my manual car. Would there be anything wrong with just connecting the two with a length of hose?
Depends on the design.
If those ports are connected to the same fluid channels in the radiator then you will need to blank them or connect together.
Many have a separate radiator built into the same structure and the auto box cooling portion is designed to have oil flowing through it. This means you can just leave it or blank as you wish.
I used that autobox cooling portion to heat diesel fuel in one conversion I did... seemed to work fine.
That’s not for coolant
That is for Automatic Transmission Fluid. Inside your radiator is another little tiny radiator that interchanges heat between the ATF and the coolant. This flows both ways: if the ATF is hotter than the coolant, e.g. Lots of heavy torque converter work, it keeps it at sane temperature. If the ATF is cooler than the coolant, it warms up the transmission (e.g. sit there for 10 minutes idling, or you are at freeway cruise in torque converter lockup for 2 hours where the automatic is doing nothing).
Your stick-shift doesn't need this; the milliseconds you spend spinning up synchros doesn't make enough heat to matter.
Its presence saves the manufacturer and parts dealers the inventory cost of stocking 2 radiators instead of 1. You can use it for anything or nothing.