Can I program new codes onto a 433mhz cloning remote?

I have a few 433mhz remotes that work with my HA. They’re cloning ones, like this.

Now the 2 different styles have different codes, but in total I only have 8 “pre-programmed” codes to work with. I was looking for a way to program/change the codes. Since I have several remotes and having them all unique would be more flexible for automations and such.

Not sure what you are asking for. If you have 8 "buttons"you will have 8 codes. If you use RFlink or other integrations you will be able to read those codes and use them. The codes should normally all be unique.
In HA though you could create more RF codes that you could use to link devices to and still use the 8 codes for essential tasks you like.
I have KAKU devices and they are not programmable but I created my own codes to control switches via HA.

Did you tried to open them so as to see if there is some dip switch to change the code ?
It could be also some jumpers or contacts to solder.