Given F. M. Alexander’s background as a reciter and actor, it is understandable why he would emphasize the art involved in teaching his technique. ‘Art’ is here used to mean a conscious and individual approach, and the opposite of a mechanistic, impersonal, and stereotyped approach.
In his first article, in 1894, Alexander wrote he did not want to be referred to as an ‘elocutionist’ but, inspired by C. T. Hartley’s book, Natural Elocution, would rather be called a ‘natural elocutionist,’ and prefers his teaching of elocution to be referred to as ‘the art of natural elocution’.
In ‘Speech Culture and Natural Elocution’ (1895) he writes that
True ease in reading and speaking, as in writing, “. . . comes from art not chance”.
(Alexander is quoting from Thomas P. Hill’s The Oratorical Trainer (1873?). The original is from An Essay on Criticism (1711), by Alexander Pope.)
In ‘Speech Culture and Natural Elocution’ (1895) he also refers to his work as the ‘the art of breathing.’ And in his ‘The Prevention and Cure of Consumption’ (1903) ‘the art of breathing’ is now the name for his technique.
In Alexander’s ‘A protest against certain assumptions’ (1910) – a criticism of Dr Scanes Spicer’s approach to breathing and voice training – he writes:
Every sound method of respiratory re-education, physical culture and voice training, must be taught as an art and can be taught only by an artist who alone is able to comprehend an art and to pass on his comprehension to the neophyte. The eye of an artist is needed to apprehend the faults in a painting or in a work of sculpture, and, above all, the defects in a human body.
To the aptitudes and intuitions of an artist, however, something more must be added before it is possible to become an efficient teacher of my principles. It is necessary to have special training in dealing with men and women and to possess that keen eye for character needed to detect and to eradicate the mental difficulties, and the vocal, respiratory and other physiological delusions which almost invariably accompany physical defects. In these things the stereotyped training of a doctor is in many ways a hindrance rather than a help, and his hard and fast methods ﬁt him as little to instruct a pupil in my methods as they would ﬁt him to teach acting, singing, or any other art.
In CCC Alexander makes a distinction between the principles of teaching and ‘the variations of a teacher’s art’:
We must learn in this connection to differentiate between the variations of a teacher’s art and the principles of the teaching technique which is being employed.
In UoS he writes that ‘the variations of a teacher’s art’ depend on the individual pupil:
It is impossible in the space at my command to put down all the details of the variations of the teacher’s art that were employed to bring my pupil to this point, for a teacher’s technique naturally varies in detail according to the particular needs and difficulties of each pupil.
The phrase ‘the variations of a teacher’s art’ has been quoted several times by teachers to indicate or to justify a variety of different ways of teaching the Alexander Technique.
Alexander’s reference to his ‘technique’ in his writings (and later to the adoption of ‘Technique’ as the name) is related to art. The English word 'technique' itself is an early 19th century loan word from French 'technique' meaning ‘formal practical details in artistic expression’. The word 'technique' is - via a Latinized adjective technicus of Ancient Greek tekhnikos 'pertaining to art' - etymologically connected to Ancient Greek tekhnē ‘art, skill, craft in work; method, system, an art, a system or method of making or doing’. 
See also Definition of the Alexander Technique.