In John 5, the cultural importance of "take up thy bed and walk" is that it takes place on the Sabbath, when Jews are not allowed to carry things according to traditional interpretations of the Torah. This results in opposition from certain Jews. The author (John) then presents sequel episodes in which Jesus explains his view of the Sabbath and his unity with the Father. This is considered blasphemous by "the Jews," who plot to put him to death.
In Matthew's and Mark's versions, the healing is controversial because Jesus has said, “child, your sins are forgiven,” prior to instructing him to "take up your mat." This is overheard by certain scribes, who object that only God can forgive sins. They accuse Jesus of blasphemy. Jesus responds by asking "Which is easier, to say to the paralytic, ‘Your sins are forgiven,’ or to say, ‘Rise, pick up your mat and walk’?" He then explains that the miracle was done so that "you may know that the Son of Man has authority to forgive sins on earth.” Taking up one's mat in these narratives has no particular significance outside of being a dramatic detail in the story.
Summary: The cultural significance of taking up the mat in John's account is that it takes place on the Sabbath, when carrying such things was thought be some to be prohibited. In Matthew and Mark, taking up the mat is a dramatic detail with no particular significance in itself. It is an incidental part of a miracle story that is an object lesson to show that Jesus has the power to forgive sins.