Linus being the second appointed after Peter is based on Irenaeus and some question if Irenaeus’ Apostolic Tradition is or was accurate.
The generally touted Catholic position about that Linus being the second pope and that he was ordained by Peter, is based, to a great degree, upon the writings of Irenaeus of Lyon.
“One Catholic scholar and priest noted:
Very little is known about Linus. St. Irenaeus of Lyons (d. 200) and the historian Eusebius of Caesarea (d. ca. 339) identified him with the companion of Paul who sent greetings from Rome to Timothy in Ephesus (2 Timothy 4:21), but Scripture Scholars are generally hesistant to do so...It should be remembered that contrary to pious Catholic belief—that monoarchical episcopal structure of church governance (also known as the monarchical episcopate, in which each diocese was headed by a single bishop) still did not exist in Rome at this time (McBrien, Richard P. Lives of the Popes: The Pontiffs from St. Peter to Benedict XVI. Harper, San Francisco, 2005 updated ed., pp. 33-34)”.
There is more and I will leave the link:
https://www.cogwriter.com/linus.htm