You can book an appointment somebody else will attend. Put their name on the booking rather than yours, and add a line in the notes saying who is paying and who is turning up. Studios hate discovering at the door that the person in front of them is not the person on the schedule.
Contact details are the part people get wrong. If you use your own phone number, every reminder goes to you and the person attending gets nothing. If you use theirs, the reminders land in the right place but the confirmation does not come to you. Decide which matters more, and tell the studio the other number in the notes.
The appointment lives in whichever account made it, so if you booked it, it is in your history and not theirs. Their Beauty Passport does not pick it up. For a one-off gift that is fine. If you are booking regularly for a partner or a parent, it is cleaner for them to have their own account and for you to pay at the appointment.
For a surprise, a gift card is usually the better instrument: they choose the date themselves, and nobody has to be woken up for a nine o'clock booking they did not agree to.