Can my wife and I share the same Expense Manager account on our phones so that we can both track our budget and expenses?
Technically, yes, you can use the same application to track the budget and expenses but you won't be able to use the same document at the same time. The software supports uploading to Dropbox and GDrive. I recommend linking the same Dropbox account to both phones, then install the application on both phones (Android recommended). This way you both will have access to the same file.
Install the application from Google Play store: https://play.google.com/store/apps/details?id=com.expensemanager&hl=en