The data (content) has always been saved in your theme files and named starting with snippets/sa-. If those files get copied over, you're set.
However, there is a way to activate saved data to metafields. These can persist across all themes. By adding atb_use_meta and atb_use_theme_based_meta to your site schema, you can then go to your site settings and turn it on. Perhaps, the reason you are noticing a change is because you had added it in the past and it is now turned off.
Add this to config/settings_schema.json if it is not already present.
,
{
"name": "SectionsAnywhere",
"settings": [
{
"type": "checkbox",
"id": "atb_use_meta",
"label": "Use Metafields for Sections Anywhere",
"info": "Save data to meta",
"default": false
},
{
"type": "checkbox",
"id": "atb_use_theme_based_meta",
"label": "Use Metafields for Current Theme Only",
"info": "With each theme data is different",
"default": false
}
]
}
A copy of the code is in your snippets/atb.liquid file.