If the search menu was used for a string property, and a data-block was
selected from the search, the value set would be an invalid name. The
property would get the modified UI string, not the proper data name set.

This problem was already once solved in rB249ccab111ac, but resurfaced
in rB937d89afba36.

Now only use the modified UI string if requires_exact_data_name is not
true.

Note: the comments in rB249ccab111ac [reg. library hints and string
properties, also that pointer properties are preferred over string
properties when dealing with IDs] still apply.