Phizzy: "Isaiah was not making a 'Messianic prophecy' at all, he is talking of an incoming King in Israel at the time which he wrote."
At least some do link this account with the future coming age of the Messiah, as shown below:
NIV Study Bible: "11:1 shoot...stump... The Messiah will grow as a shoot from that stump of David's dynasty."
11:6-9 The peace and safety of the Messianic age are reflected in the fact that little children will be unharmed as they play with formerly ferocious animals. Such conditions are a description of the future consummation of the Messianic kingdom See 2:2-4; 35:9; 65:20-25 and notes; Eze 34:25-29."
NIV Zondervan Study Bible (Edited by D.A. Carson): "11:6-9 A description of the Messianic kingdom."
The Expositor's Bible Commentary (Barker/Kohlenberger III): "The Davidic King and His Peaceful Reign (11:1-9) The Messianic theme reappears, having been out of sight since 9:7; it is taken up at precisely the point where it was left before, because the Davidic throne is implied in the reference to Jesse, David's father, and the effective zeal of the Lord Almighty in the richly empowering Spirit of the Lord."
The IVP Bible Background Commentary: "11:1-16 The Future Davidic King and His Kingdom."
The MacArthur Study Bible: "11:6-9 Such [conditions of peace] will characterize the future millennial kingdom in which the Prince of Peace (9:6) will reign."
The New Oxford Annotated Bible: "11:6-9: The messianic king (compare 9:2-7)...The Messiah will manifest the characteristics of those who were great in Israel...)."
The Jerusalem Bible: "A messianic poem describing in some detail the salient characteristics of the coming Messiah."
There are, however, other interpretations to this passage that have been published.