There are multiple ways of dealing with identifying individuals with particularly common surnames in the Spanish-speaking world.
There are athletes who will go almost exclusively by given name + surname, take for example Xabi Alonso or Dani García, who are almost never referred to as "Alonso" or "García". If you listen to Spanish cycling commentary, you will almost never hear "Rodríguez" but rather "Joaquim Rodríguez" or "Purito Rodríguez", for example. Likewise the commentators would usually refer to "Samuel Sánchez" or "Luís León Sánchez" at all times. Sometimes, especially with Luísle, they would simply call him "Luís León". This is not common in cycling, but he's a bit of a special case because of his brother being a well-known pro footballer and going by "Pedro León" for similar reasons. Other cycling examples of this kind include David López, Miguel Ángel López and Jesús Hernández, all of whom were almost invariably referenced using their given name as well as their paternal surname. Carlos Rodríguez is usually mentioned, like Purito was, in this manner on commentary.
Then there are athletes who will use both surnames in order to differentiate themselves from others. This is likewise usually typical when the first surname is common but the maternal surname is less so, in order to make themselves more readily identifiable. These are comparatively few in number, but there are a few cycling examples to call on other than those ice&fire identifies above (García Cortina and García Pierna) - such as José Ángel Gómez Marchante, José António López Gil, José Pérez Francés, José and António Gómez del Moral, Julio Alberto Pérez Cuapio and David García Dapena.
A smaller percentage - but still there - is those that choose to take the maternal surname only. The most famous example of this in recent times is former Prime Minister José Luís Rodríguez Zapatero, referred to mainly as Zapatero in reportage. Picasso is another historical example. Cyclists who do this are few in number, but there is one extremely well known example - Federico Martín Bahamontes, where there were multiple families with the surname Martín in his village so he took the maternal surname in order to be identifiable separately.
By and large, the first approach is the most common - and often arises where a rider has not selected any particular preference for using both or a maternal surname. If Carlos Rodríguez, to take a recent example, was unhappy about being identified as "Carlos Rodríguez" he could choose to be known as "Rodríguez Cano" or just "Cano", but if he wants to just be "Rodríguez", pundits and commentators will tend to mention him as "Carlos Rodríguez" frequently, regardless, as they need to have something in order to differentiate him from Cristián or Óscar, especially as all three are going to be riding a similar kind of calendar given skillsets.
With Pelayo, it would appear to me that he seems to have elected to include his maternal surname; initially it seems he may have chosen to go by the maternal surname only early in his career, but seems to be preferring going by both now - hence why his social media has the maternal name only, but he has not been correcting anybody calling him Sánchez or making open statements about the nomenclature he would prefer. Without knowing anything about him, were I a commentator I'd probably have been identifying him by saying "Pelayo Sánchez" initially, but I'd be likely to be saying "Sánchez Mayo" now.
And, as ice&fire notes, Pelayo isn't all that common as a name even in Asturias, so "Pelayo Sánchez" would still suffice to identify him anyway.