The advaitins who follow srI Adi Sankara Bagawadh pAda, are known by the name - smArthAs. Nowadys, many smArtha youngsters are unaware of this [*smArthA*] title. They refer themselves as 'Iyer' only!
VaishnavAs or vishistAdhvaitins are known as IyengArs. Dvaitins are addressed as Raojis - i.e. they attach Rao or AchAr as their surnames. As they follow srI MadhwAchArya, we call them as mAdhwAs. They also are devotees of Vishnu. But, whenever we come across the term 'vaishnavAs', we tend asscociate that only with the followers of RamAnujA's siddhAntha. As mAdhwa's have 'AchAr' as their surname, srI vaishnavAs also have 'AchAryAr' as surname. They have two variations in this, such as 'Raja gOpAla IyEngAr' and 'Raja gOpAlAchAryAr'.
As the mAdhwAs are widely known based on the name of srI MadhwAchArya, the followers of srI Sankara or srI Ramanuja are not known to be referred to by the names of their respective achAryAs. The term 'SankarAdvaitin' may be present in some English philosophy book; may float in lectures, but not in vogue in day-to-day life. In practice, we commonly have dvaitins as Raos, vishistAdvaitins as IyengArs and advaitins as Iyers.
Generally, smArthAs alone have shAsthry, sharma etc. as their surname. Even those who have 'dhIkshidhar' as their surname are invariably a small segment of smArthAs in our region. [*PeriyavA is referring to Tamil Nadu*]. The title of dhIkshithar is awarded to one who conducts yagAs and to his family. (The case of dhIkshidhar of Chidhambaram is quite different). In general, advaitins consider themselves as those belonging to Iyer subsect. We, thus, conclude based on the surname in vogue. [* ie Iyer].
This generation doesn't use the subsect as surname. This has become a taboo! Even in the previous generation, others didn't employ the subsect name as much as Raojis did. If we go a generation or two backwards, we can conclude that we are grandson(s) of either an Iyer or IyengAr or a Raoji. Even those who had 'AchAr' as surname, did not refer to themselves as those belonging to 'AchAr subsect' - they referred themselves only as Raos or mAdhwAs. Similarly, vaishnavAs with 'AchAryAr' surname, don't refer themselves as belonging to AchAryAr subsect - but consider themselves as IyengArs only! [*We*] smArthAs refer ourselves as Iyers: not as those belonging to sharmA subsect or shAsthry subsect.
As much as followers of srI Sankara Bhagawad pAda are not aware of their being smArthAs, [*they*] are equally unaware of another important fact! The fact that [*Adi*] AchArya hasn't identified any particular deity as an exclusively chosen one for them and instead advocated to view all of them equally [*with equal veneration*] is not known even to many advaitins.
Thinking that [*Adi*] AchArya has chosen Parama Sivan as their 'upAsanA mUrthy', they identify themselves as Saivaites. There appears to be a reason for this line of thinking.
Both dvaitins and vishistAdvaitins are followers (upAsakAs) of Vishnu. It appears that the smArthAs of today who happened to be advaitins, consider themselves to be saivaites as they differ from the other two subsects. But, this reasoning is wrong.
Iyer, IyengAr or Rao are not sanskrit names. Hence, when an IyengAr refers to his subsect, he introduces himself as a vaishnava; Raoji does as a mAdhwA. There is nothing wrong in their [*such*] identification. But, only this Iyer who happened to be an advaitin, refers to himself as a 'saivaite'. Even others seem to hold this wrong notion!
Even I'm considered to be a saivaite AchAryA by many. A professor from Japan (Hajme Nakamura) had come to me. He said, "I've studied [*Adi*] AchAryAl's sUtra bhAshyam, gIta bhAshyAm etc. In all of them, only advaita is elaborated but no mention had been made at all about Siva" and asked me "How is that you remain a Saivaite ?"
I asked him back, "On what basis do you consider me a saivaite?". He replied,"You are wearing only vibhUthi? You are doing Shiva pUja (Chandhra moUlIswara pUja). All other ShankarAchAryAs are also happened to be saivaites like this. When [*Adi*] AchAryA hasn't found any bhEdha between Shiva and Vishnu, why is that you and those of other Shankara MathAs wear saivaite symbols and do Shiva pUja?"
I answered him [*thus*].