Jurisdictions: england-and-wales and eu
Article 6(1) of the UK GDPR (and also of the GDPR in relation to EU law) provides that it is unlawful to process personal data unless the data controller can establish one of the 6 lawful bases set out in that Article. Article 4(1) defines what is meant by "personal data". Article 4(2) defines "processing". This is a very wide definition which includes things such as collecting, recording, using, and disclosure.
By accepting the appointment from your wife, the doctor's surgery has to collect and record things such as your name, the fact that you have an appointment booked on a particular date, and perhaps some other details such as the subject matter of your appointment and your date of birth. Depending on the surgery's processes (i.e. whether or not they accept appointments from non-existing patients), the fact that they accept the appointment at all might disclose to your wife that you are already a patient with that surgery. These are all things which would fall under the definition of personal data.
Aside from privacy law, there are other potential legal issues to accepting the appointment from a third party. For example, at a private clinic which charges fees, a contract will be formed at the moment that the surgery agrees to see you. If your wife wasn't authorised to act on your behalf, that could be problematic.
Even if it weren't the case that the law prevents the surgery from taking appointments from third parties, there is also nothing which requires them to do so by default, and a person is entitled to refuse to do something which they are not required to do.
These issues can be overcome by authorising your wife to act on your behalf. Doing so also provides the service provider with a lawful basis under the GDPR (consent). In england-and-wales there are two common ways of doing this: agency, and power of attorney.
With agency, you appoint someone to be your agent. For anyone dealing with the agent, it is mostly equivalent to dealing with you. For example, the agent can enter into a contract with someone and it is as if you had entered into that contract. There is no prescribed method of appointing someone as your agent, and you don't have to use the word "agent"; a verbal agreement between you and your wife can be sufficient. A service provider will typically want you to write them a letter to evidence the fact that you have authorised the agent to act on your behalf. Some might want you to do this on their own form. Others might accept a verbal confirmation from you which they will then keep a note of. You would have to ask the service provider what their process is and then follow that.
A power of attorney is a formal legal instrument. There are different types; the simplest and most common is an ordinary power of attorney. This is issued under Section 10 of the Powers of Attorney Act 1971 and must be written in the way described in that Section. An ordinary power of attorney "operate[s] to confer [on] the donee of the power [...] authority to do on behalf of the donor anything which he can lawfully do by an attorney". Although this is a formal document, it is relatively easy to produce one. You simply need to make sure it is worded, signed, and witnessed correctly.
An ordinary power of attorney should usually be sufficient for doing routine things such as booking appointments, and service providers should not refuse instructions from a donee, provided that they are satisfied as to the validity of the document. However, an ordinary power of attorney cannot be used where the donor has lost capacity (e.g. is in a coma). In that scenario you need a lasting power of attorney under Section 9 of the Mental Capacity Act 2005. This is a more formal document than an ordinary power of attorney. There are more restrictions on what it can include, and it must be registered with the Office of the Public Guardian (for a fee) before it becomes valid. If drafted correctly, this would allow a third party to make medical decisions (including booking appointments) on your behalf even in cases where you lack the capacity to do so yourself.
Warning: powers of attorney are very powerful, and you should exercise extreme care when drafting one. For example, they are capable of conferring powers to sell your assets and spend money. A lasting power of attorney can confer the power to discontinue life sustaining treatment while you lack capacity. Usually you will want to lock down the wording so that its effect is limited in the way you intend, and it's advisable to use a lawyer for this unless you know what you are doing.
I am not familiar with the law in Switzerland but a quick Google search suggests that they have the concept of a power of attorney there.
