Is Thenkalai Iyengar sect really Brahmin? Why do they claim as Sri Vaishnava Brahmins?
Yes, Thenkalai Iyengars are really brahmins.
Thenkalai or Tenkalai is one of the traditions in Sri Vaishnava tradition which subscribes to teachings of Pillai Lokacharya and Manavala Mahamuni. The other being Vadakalai follows Vedanta Desikan. There are mainly 18 differences between these two sects in matters like Kaivalya Moksha, how to wear the namam etc.,
When it comes to Iyengar, it is the name given to group of Brahmins who followed Sri Vaishnavism due to advent of Ramanujacharya or from the time of Natha Muni. They perform upanayanam, wear yajnopaveetam and perform all the rituals done by Brahmins because they are Brahmins by birth.
They claim as Sri Vaishnava Brahmins because they worship Vishnu as supreme (hence Vaishnava) and take initiation from Sri tradition. Their varna by birth is Brahmin. Hence, they also claim as Sri Vaishnava Brahmins. This distinction helps because there are Smarta, Shaiva and many other types of Brahmins who are Brahmins by their birth varna but their philosophy and faith is different. Adding the name Iyengar or Sri Vaishnava before their doesn't remove them from Brahmins because they are added for classification.
Brahmin denotes the varna they are born. Iyengar denotes that they follow Ramanujacharya's philosophy i.e., Vishishtadvaita (non-dualism) and faith Sri Vaishnava. In the same way, Sri Vaishnava denotes their faith and Thenkalai shows their affiliation towards theological differences within Sri Vaishnava tradition.
As the time gone by, the name Iyengar became a surname for families who once followed Sri Vaishnava in states Tamil Nadu and Karnataka. Now, there are Sri Vaishnavas following Tenkalai tradition who are not Iyengars out side these states. At the same time, who have the name Iyengar who are not initiated into Sri Vaishnava and do not follow Ramanujacharya's philosophy.
Yes, the Srivaishnava Guruparamparas (both on Thenkalai side and Vadakalai side) mention intermarriage between the two groups which wouldn't happen if they weren't both Brahmin. Keep in mind, Vedanta Desika and Periyavachan Pillai both disallowed intermarriage between Srivaishnavas if they were not both from the same varNa.
There certainly are non-Brahmanas adherents of Ramanuja Sampradaya who sport Tenkalai or Vadakalai forehead marks and are initiated into the respective sampradayas. But presumably you are not referring to them.
The Tenkalai Sri Vaishnava position represents Sri Vaishnavam as presented by Ramanuja and 70 of his 74 Simhasanadipati Peethams. Formally called the Ora:n Vazhi: paramparai, it focuses its message on A:zhwa:r Arulicche:yal supported by Ve:dam as it is revealed in itiha:sam. This is stated in Sri: Vachana Bhu:shanam of Swamy Pillai Lokachariar.
Since such sources are open to all, the Tenkalai community welcomes people of all castes into the fold. With the exception of temple worship and Ve:dic ritual, which is limited to Sri: Vai:shnava Brahmins, all other activities are open to all castes and communities.
As one studied in the tradition, I speak with certainty how the title of Brahmin is not used in general conversation. We prefer instead Ramanuja Dasa.