This question/problem came up in the Backdrop CMS issue queue. The problem arose when someone was porting a D7 module that stored configurations at the root level of the field instance definition. Those custom configurations did not get saved. Backdrop doesn't allow custom configurations at the root level of the instance - it restricts that level to specific, pre-defined keys, and ignores everything else.
The solution is to modify the custom module to store its custom configurations inside the
settings element of the instance array. Then it will get saved. For example, you could store your settings like so:
$instance['settings']['my_module'] = array('setting_1' => 'asdf', 'settings_2' => 'zxcv');