My referee is a pioneer of my university in my home country. I am applying for PhD in Australia. My referee has signed the letter of recommendation in hardcopy form but he hasn't put a stamp of his name on it saying that letter head needs no stamp. I have mentioned on letter head that he is Professor and Trustee. I have asked electrical department to put department stamp on it and they agreed. In this scenario is it likely that my chosen university in Australia may question why the pioneer and professor hasn't put his own stamp? Is it likely that the Australian university would contact my referee for it? Note that I am applying for a highly competitive scholarship in Australia.
Australian universities do not use or expect stamps. However, they may be familiar with the practices of universities in Asia and notice inconsistencies. I suggest this is unimportant.
I'm at a UK University and we don't have individual stamps, nor do we put any official stamp on reference letters. I've written dozens of recommendation letters for students applying to Australian Universities and none have had a stamp. Almost all of those students were admitted, many with significant funding. Listen to your Prof, you're fine.