Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
What some people do when they hear the name of the Prophet (Sallahu Alaihi Wa Sallam) being said during the Adhan, i.e. kissing the nails of their thumbs, and has declared this act as Mustahab. However, there is no authentic evidence for it.
Imam Dailmi quotes this hadith of Abu Bakr Siddiq (Radi allahu anhu), "When Abu Bakr Siddiq (Radhiallahu Taala Anhu) heard the Muazzin say he read and kissed the segmented part of the fingers of Shahadah from the inside and placed them on his eyes. Upon this the Beloved Prophet of Allah said,"Whoever does like my friend has done, for him my intercession (Shafa'ah) will become halal.
Allama Sakhawi has stated that this Hadith is unauthentic and incorrect. (Al-Maqasidul Hasanah by Imam Allama Shamsudeen Sakhawi Rahmatullahi alaih pg.384 Hadith1021).
In the book of firdaous this Hadith is found. But in regards to this Shah Abdul Azeez Dihlawi (RA) wrote that this book contains many fabricated (Mawdoo') narrations. Mawdoo' are those narrations, which the prophet (Sallahu Alaihi Wa Sallam) had never said but someone imposed a lie upon the messenger of Allah (Sallahu Alaihi Wa Sallam).
This narration could also be found in the books of Kanzul-Ibaad and Fatawa Sufiya. But Allamah (Ibn Abideen) As-Shami (RA) wrote in his Raddul Mukhtar: “Fatawa Sufiyah is an unreliable book, giving a Fatawa upon this book is not permitted”.
Then Allahmah Ibn Abideen (As-Shami) RA further discusses about this narration: “After Jarahee (RA) had a long discussion in regards to this Masala (issue) he wrote: “There is absolutely no (authentic) Marfoo Hadith found in regards to the kissing of the thumb which could establish this to be a Masnoon or a Mustahab act. (Fatawa Shami Vol 1 p467)
Many masters of Hadith have classified the narration to be fabricated i.e Hafiz Ibn Taahir al-Fattaany (Tazkiratul Mawdhoo'aat pg.34).
Also, Mullaa Alee al-Qaaree refers to this hadeeth in his 'Mawdoo`aatal-Kubraa' [no. 435].
Therefore, one cannot conclude that this is a Sunnah or Mustahab based on this unreliable narration.
However, It would be disrespectful to do anything against the Sunnah of Rasulullah (sallallaahu alayhi wasallam) and To carry out such an act as a Mustahab or Sunnah is Bid’ah (innovation) and a sin.