'Str' object is not callable error for integer values

Look at the error message:

'str' object is not callable' error at this line -->

av=date(day, month, year).strftime('%d %B %Y')

You are calling two functions here: date and strftime so one of those is wrong.
Break the line in two and see if that helps:

xav=date(day, month, year)
av= xav.strftime('%d %B %Y')

Now you get the error on the first one:

xav=date(day, month, year)

So you at the surrounding code:

def date_fun(date):
  
  try:
    dt = parse(date)
...
    xav=date(day, month, year)

date is a parameter to the date_fun function, which means it overrides the function of the same name, so when you try to build a new date, it thinks you mean to use the variable.
Change the parameter name, and the error will go away:

def date_fun(datestr):
  
  try:
    dt = parse(datestr)
    dt = dt.strftime('%d/%m/%Y')
    dmy_split=re.split('[- / ]', dt)
   
    day=int(dmy_split[0])
    month=int(dmy_split[1])
    year=int(dmy_split[2])
    av=date(day, month, year).strftime('%d %B %Y')

    if (day>=1 and day<=12 and month>=1 and month<=12 or len(year)==2):
                                                 print("Ambiguous; likely dates{}")                
    else:
                            print ("True,{}".format(av))                              

  except ValueError:        
                                                        
                                                     print("False")

And please, sort out your indentation!

You should expect to get syntax errors every day, probably many times a day while you are coding - we all do regardless of how much experience we have! Sometimes, we misspell a variable, or a keyword; sometimes we forget to close a string or a code block. Sometimes the cat walks over your keyboard and types something really weird. Sometimes we just forget how many parameters a method call needs.

We all make mistakes.

And because we all do it, we all have to fix syntax errors - and it's a lot quicker to learn how and fix them yourself than to wait for someone else to fix them for you! So invest a little time in learning how to read error messages, and how to interpret your code as written in the light of what the compiler is telling you is wrong - it really is trying to be helpful!

So read this: How to Write Code to Solve a Problem, A Beginner's Guide Part 2: Syntax Errors[^] - it should help you next time you get a compilation error!