Most, if not all, of the sampradayas of Vaishnavism, Saivism, Shaktism etc. have an initiation or diksha process for even common folks to be part of the particular sampradaya they are attracted to/meant for. Now, at least in the southern states to my knowledge, there are many smartha families that claim to be followers/devotees of one of the 2 major mutts associated with Adi Sankaracharya. My question is how do people/families in general claim lineage or association with a particular mutt if there was no initiation done? How to they accept either of the Sankaracharyas as their Guru? Because, to my knowledge, the sankara sampradaya does initiation process only for sanyaasis. Has Adi Sankaracharya given any instructions in this regard?
It has been close to a month since I asked this question but unfortunately, no one has an answer. Therefore, I am going to attempt an answer based on what I have read and heard with references I could gather.
As far as I have seen, from my direct experience as part of the 'iyer' community, there is no separate diksha or initiation process to become part of the Shankara Sampradaya for common folk. Nor is there a practice of sticking to a lineage of family acharya, aside of the madathipathis (again, only with reference to Sringeri and Kanchi, don't know about the other mathams).
This page describes the process of upanayanam as "initiation" but it is certainly not in the same sense of the diksha or samaashrayanam. Also, another difference that I have noticed is that only sanyaasis are eligible for such diksha in Shankara sampradaya. Or should I say, one is required to take up sanyaasa along with the diksha. Not so in the case of, say Srivaishnava sampradayas as can be seen in the answer by Shriman Keshav Srinivasan to this question.
The most interesting link I had come across while on the search for this topic is this, on the worldly departure of Brahmasri C V Kamakoti Shastrigal, Srividya Sthanigam of Sri Kanchi Kamakshi Temple.
Here, it is clearly mentioned that he had received some form of diksha from one Brahmasri Chidanandanatha (Sri 'Sir' Subrahmanya Iyer) in Srividya upasana. This also indicates that being considered a 'smartha', subscribing to 'advaita' and following what is clearly a 'shakta' practice are all totally disconnected. One more example is given here.
An ardent worshipper of Amba in the form of Sri Bala Tripurasundari, his Diksha name was Sri Balanandanatha. He has thousands of loving disciples who have received invaluable guidance from him in Srividya Upasana.
The article also mentions that he had come in touch with Sri Chandrashekharendra Saraswati Mahaswamigal of Kumbhakonam Mutt. The article also states this:
Sri Shastrigal, apart from being a scholar par excellence, was a great Mantra Siddha who helped hundreds with his mantra Siddhi. He was very close to H H Paramacharya Sri Sri Chandrashekharendra Saraswati Mahaswamigal and remained his most trusted Shishya.
Even with the usage of the word sishya here, it can be safely assumed that there was no diksha link between them, because history tells us that only 2 people were given diksha by Paramacharya. This point also has an interesting fact for those people who can grasp it.
So, to conclude, there is no known process for diksha for non-sanyaasis in the Shankara sampradaya. The connection of families to particular Shankara mathams was mostly through familial traditions or through attraction to the individual mathams.