Having been that receptionist a few times* I can tell you about it from the other side.
I'm UK based and was working for NHS GP surgeries. Receptionists are bound by all the same patient confidentiality rules. They're just receptionists, no special training apart from the list of reasons that get you through regardless, the doctors make any decisions on priority based on the reason given.
From your description of the conversation you were offered an appointment on the same day, that would class as an urgent appointment and require a reason to be given.
You are permitted to say it's personal or private. The receptionists don't want more than one or two words to describe it, it's not a very big text box. We certainly don't want the extensive details that too many people gave us as soon as they were given the opportunity.
Asking a reason for the appointment normally only applies to urgent appointments, special appointments and nurse appointments. Every so often a doctor will ask for a reason on every appointment but it never sticks as it takes too long.
For a doctor the special appointments are normally things like baby and maternity checks that take longer. If you're booking multiple slots for a single appointment the doctors are going to want to know why, a reason is always required for this.
Nurse appointments often require equipment such as dressings or vaccinations that may not be stored in their own room fridge. Putting a reason on the appointment allows the nurse to have everything ready before calling you in. Nurses will not normally call a patient in without knowing why they're there, if they have to leave a patient in the room they'll need to call another member of staff in. A patient should never be left alone in a clinical room.
Urgent appointments can be prioritised based on the reason given. "Personal" will come below "Chest Pain" or "Baby having asthma attack" for example. Some reasons also allow overbooking of appointments, there's a set of reasons which will allow you to book an appointment on the same session even if all the listed appointments are filled. Beyond that list, the receptionist doesn't care what your reason for wanting the appointment is, they must not attempt to make judgement on the urgency of your condition, only a clinician can do that.
The reason the receptionist asks at the end of booking the appointment is simply because that's when the dialog box comes up asking for it.
*It's a tough job but not a thankless one, around Christmas we lived on chocolate provided by the regular patients. Soooo much chocolate, but more is always appreciated.