How can I identify which batch a sales/pay invoice is in?

You can see the batch id when you go into Invoice Edit mode, so for example for a sales invoice go to Sales Invoices > Invoice Edit, enter the invoice number and the batch id will be visible at the top of the screen:

The equivalent option is available for Purchase Invoices via the Invoice Edit option. The batch id is also visible on the timesheet - if you select them from Timesheet Display it is visible here: