The BBC wildlife presenter, who found fame in the 1970s as a member of comic trio The Goodies alongside Tim Brooke-Taylor and Garden, died on Saturday, aged 85, according to his agent.
Tributes have poured in for Oddie, who has been remembered for his humour as well as passion for the natural world.
Speaking on BBC's Radio 4, Garden said: "We were very different personalities. I think Bill was much more of a rebel than I was, and was much tougher in many ways. We laughed at the same things. Anything that happened to the Goodies made us laugh.
"He was the one who would challenge the status quo.
"He was the one who wouldn't wear a tie in the restaurants and he was the one who stood up for himself."
He recalled how Oddie was a "great giggler" and that he would often start giggling in the middle of a scene, which Garden said was the "seal of approval".
"This is a terribly sad time," he added, "but as you probably gathered, I have so many very happy times to look back on and remember and to enjoy. Now we can look back at the memories and and we can rejoice in what what a wonderful chap he was."
British TV nature presenter Chris Packham described Oddie as an "utter joy to be with".
The naturalist, 65, is best known for hosting wildlife program for the BBC such as Springwatch - having replaced Oddie as the face of the show in 2009.
Packham was among the famous faces flocking to social media to remember Oddie, sharing a picture of him alongside a lengthy and emotional message.
He wrote: "A renegade polymath with binoculars and a beard, Bill was invariably funny and occasionally furious.
"He loved birds and loathed those who harmed them. He was delightfully unpredictable, sparky, smart and at times a bit spiky.
"Critically he really knew his stuff, and his passion was true - he didn't just pick up his binoculars on telly, he went birding whenever he could.
"So he was genuine and that's why we loved him, he never needed a script, and took scant interest in any he was given.
"He didn't like rules and often neglected to play by them. This may have frustrated the serious and the worthy but it always amused his audience – he was an utter joy to be with."
Others to pay their respects include Monty Python star Eric Idle, who first met Oddie while attending Cambridge University, and they both rose to fame as actors and comics in the 1960s.
The Grammy-winner, 83, wrote on X: "I'm very sad Bill Oddie has gone. He was sick for some time and I wrote to him.
"He auditioned me for The Pembroke Revue in 1963 and I stepped into his roles for The Edinburgh Festival when the Cambridge Footlights went to the West End.
"We watched England win the World Cup together at Wembley in 66. He was a very fine songwriter and I learned a lot from him."
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