The actor and comedian, who in 2017 won damages himself from the publisher of the Daily Mirror over illegal phone-hacking, says he has “great sympathy” with the prince.
However, he said it was important to remember that it isn’t just those in the public eye who have been affected.
Coogan spoke about the issue at the premiere of his new film Greed, a satire about a billionaire high-street fashion mogul.
Asked if he believes Prince Harry has done the right thing, the star said: “Yes I do, and I think it’s important.
“Independent accountability in the press is still a very live issue, a very important issue.
“I’ve got great sympathy with him. But we’ve got to remember it’s not just royals and celebrities who are affected by press abuse, it’s people who have no money and don’t have the means to take legal action…
“I hope he shines a light on an issue but it’s important to remember it doesn’t just affect people like Prince Harry, it affects ‘mere mortals’ who don’t have the same resources too.”
Buckingham Palace has said that Harry has filed claims at the High Court over the alleged interception of voicemail messages. It is understood the allegations relate to historical incidents.
The phone-hacking scandal led to the closure of the News Of The World, one of the UK’s most popular newspapers at the time, in 2011, as well as the conviction of key journalists, a huge police investigation and the Leveson inquiry examining the culture, practice and ethics of the press.
Coogan, who gave evidence, said he had “always supported a second Leveson inquiry”.
“I’m hoping that will come along,” he said. “We’ve been ploughing at it for quite a long time. As we know, politics is quite a precarious thing at the moment – we never know who’s going to be in power in a couple of years time.”
Coogan also spoke about his character in Greed, Sir Richard McCreadie – known as “McGreedy” – who isn’t dissimilar to Arcadia Group owner Sir Philip Green.
He described the businessman as a “gift for dramatists and comedy”, but said he also drew from other inspirations.
“In some ways it’s unfair just to point the finger at Philip Green because he’s the most, if you like, theatrical – he did court publicity in certain ways and in some ways he’s something of a pantomime villain for a lot of people when it comes to this subject.
“But of course there are a lot of other people, there are [high street fashion] companies… that are so rich they could double the wages of the people they pay in these garment factories and they’d still have a very comfortable profit margin, and still have their super yachts and everything else. The bottom line is it’s about greed.
He added: “Philip Green’s sort of a gift for dramatists and for comedy. We sort of took some elements of him but part of it is invention as well. There are a lot of direct quotes from him but obviously I don’t look like Philip Green so there was some artistic licence in there as well.”
The film features his character’s lavish 60th birthday celebrations on the Greek Island of Mykonos, but Coogan said his celebrity partying days were behind him.
Asked about the wildest party he had been to, he said: “It was probably about 20 years ago and I can’t remember, which is probably a good thing.”
Speaking about his famous night out with Liam Gallagher – which the Oasis star has claimed ended with the pair waking up in the same bed together – Coogan said: “That was a crazy drunk night.
“Each time he repeats that story and I repeat it, it gets more and more exaggerated. It was a very enjoyable but drunken night in Ireland and we were young and foolish and now we’re old and foolish.”
Greed is out in UK cinemas on 21 February 2020.