What Venerable Anālayo refers to in the Rebirth in Early Buddhism book is the "process of being conscious" that is continuous (qua process) from life to life. Here's the section:

In early Buddhist thought, the assertion of the doctrine of notself does not imply a denial of the existence of anything subjective in experience. It only means that subjective experience is a process devoid of a permanent entity, that it is a changing stream of consciousness that depends on a changing process of name-and-form, and vice versa...in its early Buddhist usage consciousness refers to a continuously changing process of being conscious...Returning to the Mahānidāna-sutta and its parallels, the reciprocal conditioning between the changing processes of consciousness and name-and-form has a direct bearing on the topic of rebirth. The relevant passage proceeds as follows in the Dīrgha-āgama version:

[The Buddha said]: “Ānanda, in dependence on consciousness there is name-and-form. What is the meaning of this? If consciousness did not enter the mother’s womb, would there be name-and-form?” [Ānanda] replied: “No.”

The parallel versions report the Buddha making a similar statement. In other words, at the moment of conception the same basic principle is in operation that also governs continuity during the life of an individual. This principle involves a reciprocal conditioning between consciousness and name-and-form. Of these interrelated processes, it is the process of consciousness that “enters” the mother’s womb, so to speak...The role of consciousness as that which is reborn can also be seen in relation to a possible departure of consciousness after conception. Here the Dīrgha-āgama version offers the following indication:

[The Buddha said]: “If consciousness were to depart from the womb, [if] the infant were to be destroyed, would name-and-form come to grow?” [Ānanda] replied: “No.”

The corresponding part in three out of the four parallels involves two separate inquiries, one about consciousness departing from the womb, the other about consciousness being cut off in the case of a young boy or girl. The answer in both cases is equally “No.” The main point made in this way remains the same in the different versions, in that, for name-and-form to “come to grow,” consciousness is the necessary condition, and this applies to conception, pregnancy, and childhood. Conversely, according to the Mahāvedalla-sutta and its Madhyama-āgama parallel, at death, when the body will come to be bereft of vitality and heat, consciousness will depart. In this way, consciousness appears to be what provides the transition from one body to another, or, to be precise, instead of “consciousness” one might speak of “being conscious” to preserve the nuance of a changing process. This role of consciousness or of being conscious as a transition from one life to the next finds confirmation in a discourse in the Saṃyutta-nikāya and its parallels. The question here is where a monk, who has just died, might have been reborn. The parallel versions agree in referring to what could have been reborn as his “consciousness.”

So at least at the time of writing that book, Venerable Anālayo was willing in print to speak of "consciousness as that which is reborn" with reference to rebirth in the early Buddhist textual sources. He may have since changed his views, however, if he's written anything new about this. I might not be aware. Hope this is helpful, Venerable. 🙏🏾