Yes, in most setups you can save customer details in the software and reuse them, so your staff are not asking the same questions at every booking. Travel agency software normally keeps a customer profile where you can store the name exactly as it appears on the passport, date of birth, nationality, passport number and expiry date, contact details and any frequent flyer or loyalty numbers. When the same person books again, your staff pull the profile into the booking instead of typing it all out fresh, which also cuts down on the name spelling mistakes that are awkward to correct once a ticket is issued.
What happens to those details after that depends on the supplier and the API behind the booking. Some airlines and hotel suppliers want passport information at the time of booking and will accept it straight from the system, while others only need it at check in or at the airport, so the field may simply sit in your records rather than being passed on. Which fields are available to save, and whether they flow through automatically, can also vary by plan, so it is worth confirming before you build a process around it.
There is a responsibility side to this as well. Passport and date of birth information is personal data, and you are the one holding it, so you should only keep what you genuinely need, tell your customers you are storing it, and use the staff permission settings so that not everyone in the office can see or export it.
The sensible next step is to open a customer profile in your own account, see exactly which fields are offered, and ask your account contact whether those details are sent to your suppliers at booking time or only kept on your side.