Two banners I've designed in Indesign (to be printed) will be placed next to each other, and there is an image that continues from the first to the second banner.
How can I make them join together when I have a 10mm bleed, which will cut off the image at the edges and therefore they won't match?
The bleed is the "extra" of your design, not the design itself.
So what you need to do is use the top part of the second banner as your bottom bleed for the top banner.
Since the bleed zone is meant to be trimmed or disappear, you can't use parts of the design that you want to be visible in that zone.
I uploaded an example to make my answer easier to visualize:
