It is regular.
Lets see the strings in L
= { a, b, aa, ab, ba, bb, aaa, ..... } (When w is ϵ, wxw generates all these strings and hence we don't need to consider any other case for w)
= Σ* - {ϵ}
As we have no restriction on the length of x, we can leave 1st and last symbol as it is and consider the rest part as x.
eg : w=bab x=bb
wxw = babbbbab; we may assume x = abbbba so we can read it as : bxb which is finite.