I have seen this happening already at two different companies. And I'm also doing it as well. Particularly for search indexes where there's no risk of data loss.
Data loss is the obvious concern. Are you perhaps reusing parts of sqlite or postgres?
There'd be no benefit. They're good general purpose databases but for any specific workload you can do significantly better on your own. For search in particular even if you use Postgres you wouldn't want to use Postgres FTS you'd want to at least use Tiger Data's extension.