#Mathematica 11.x, 74 bytes
Mathematica 11.x, 74 bytes
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&
where, throughout, # represents the input string, and
StringCases[#,<pattern>]
finds substrings of the input string matching the pattern
StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")
This pattern requires matches, x, must start at the start of the string and must satisfy the condition that (1) the match is not the whole input string and (2) if we replace occurrences of the match in the input string with the empty string we obtain the empty string. Finally, comparing the list of matches to the empty list,
{}!=
is True if the list of matches is nonempty and False if the list of matches is empty.
Test cases:
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&["aa"]
(* True *)
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&["aaa"]
(* True *)
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&["abcabc"]
(* True *)
and
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&["aba"]
(* False *)
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&["ababa"]
(* False *)
{}!=StringCases[#,StartOfString~~x__/;(x!=#&&StringReplace[#,x->""]=="")]&["weqweqweqweqweqw"]
(* False *)
