Get current table name trigger was fired on
Is there any way I can obtain inside a trigger the name of the table on which the trigger is defined?

No, there is no such capability in any of the current (as of year 2020) Db2 versions, for reasons that I won't go into. For a comparison, consider why a C function cannot know its name or the name of the function that called it.
For each trigger that you write you must know the corresponding table name when you write it, simply for the fact that it is part of the trigger definition:
CREATE TRIGGER &lt;name&gt; ... ON &lt;table name&gt;

and it is trivial to also add that name, as a literal, to the INSERT statement.
If you face a tedious task of creating an inordinate amount of nearly identical triggers, use your favourite scripting language and one of the template engines available for it to define a trigger template and generate DDL statements for all your triggers from it.
Alternatively, consider using the Db2 audit facility, which is perfectly capable of doing what your triggers are meant to do.