Maybe you are misunderstanding the term analogue in analogue amplifier. In recent times the word has become to mean the opposite of digital that is a continuous function instead of a discreet function. But that is not what it really means.
The current flowing from collector to emitter in a transistor is an analogue of the current flowing in the base to emitter. That is it is "like" the current but subject to some function, the simplest of which is gain, the multiplication by a constant.
Rather than a relay being an amplifier, it is in fact the amplifier that is acting like a switch when you are dealing with digital signals.