That includes the Siege of Mandalore, in which Anakin would have participated had it not been for the Chancellor's kidnapping. But their story began years before that, Darth Maul and Anakin's paths briefly crossed in Tatooine and later in Naboo.
Obi-Wan Kenobi was Darth Maul's nemesis in Star Wars, but there was a time when a young Anakin Skywalker dueled and defeated the Sith Lord. Before the Clone Wars forced them into military positions, Star Wars' Jedi filled many roles for the Republic.
So, with Maul not really caring about Anakin, he thought it was a safe bet to escape. Unfortunately, from here onward, no confrontation between the two has ever been seen in any Star Wars media. However, that doesn't mean it never happened.
In a non-canon story from the pages of Star Wars Tales #9, Darth Maul was resurrected to fight Darth Vader to the death. Ultimately, Darth Vader was named the victor for a pitch-black reason: He hates himself more. In the main Star Wars canon, Darth Vader and Darth Maul never met, let alone fought.
Yep. Darth Maul didn't really say or show straight up that he "feared" Darth Vader, but he openly stated his belief that he could not defeat Darth Vader without the help of the Jedi, therefore on his own.
By the time of Episode 3; Anakin is in his prime, and already one of the Jedi Order's most powerful “masters” (although technically still a knight), just behind Obi-Wan, Windu, and Yoda. Maul wouldn't stand a chance. Anakin stomps.
Ahsoka asks what he wants with Anakin Skywalker. Maul says that Skywalker is the key to everything. When Tano asks if it is to bring balance to the Force, Maul replies that Skywalker is the key to destroy and that he has long been groomed for his role as his master's new apprentice.
Later, Maul orchestrated the siege of Mandalore as a means to draw Anakin Skywalker and Obi-Wan Kenobi to him with plans to kill Skywalker and deprive Darth Sidious of his new apprentice.
Darth Maul could have killed Anakin Skywalker at least twice and therefore prevented Darth Vader from being created, but that does not mean he would have been powerful enough to stop Darth Sidious' rise to power.
The fight was orchestrated by a darksider cult called the Prophets of the Dark Side, who wanted Vader killed and replaced with Maul as Darth Sidious' only true apprentice, because they felt the former was "tainted" by his Jedi past and not worthy of being a Sith.
Having been trained to become Sith Lords, both Maul and Dooku sought to gain more power under Sidious while simultaneously looking to overthrow him. Toward the end of the Clone Wars, Dooku and Maul did actually meet each other in canon in the Son of Dathomir comics by Jeremy Barlow.
Darth Vader's true identity as Anakin Skywalker was a highly guarded secret, known only to a select few individuals including Palpatine, Obi-Wan Kenobi, Luke, Han, Leia, R2-D2, C-3PO, Master Yoda, Bail Organa, Ahsoka Tano, Darth Maul, Grand Moff Tarkin, Grand Admiral Thrawn, Jocasta Nu, Ferren Barr, Verla, Reva, ...
Finally, Darth Maul himself admitted he wasn't capable of facing Vader alone in Star Wars Rebels. And while the story in question is no longer part of the official Star Wars canon, Darth Vader fought and defeated Maul in comic Star Wars Tales #9.
Ren was good but Rey was able to beat him every time they fought. However, he wasn't even fully trained and even he would have had a massive problem fighting Maul. Even in his older age, Maul was still way more skilled than anyone Rey faced and that would be the end for her. Maul would just be too much for Rey.
It's already been shown in the source material… more than once… that Darth Maul is not on Darth Sidious's level. Every time he dueled Sidious as his apprentice (in Legends) he got easily defeated. In the show Star Wars: The Clone Wars, Palpatine takes on Darth Maul and Savage Opress simultaneously.
The real reason for Obi-Wan's victory, though, may have been that he was the better fighter. While Darth Maul's moves were flashy, Obi-Wan eventually proved himself as one of the best duelists in the galaxy, and his fight with Maul was just the awakening of his skills.
Though seemingly killed by Obi-Wan Kenobi at the end of the film, Darth Maul returned in the 2008 animated series Star Wars: The Clone Wars (voiced by Sam Witwer).
Sidious revealed to Maul that he did not intend to kill his former apprentice, as he still had use for Maul; it was Sidious' hope that Maul would lead the Sith to Mother Talzin, Maul's mother and the leader of the Nightsisters, so she could be eliminated as a potential threat to the eventual Sith rule of the galaxy.
Chastened by his former Master on the principle of the Rule of Two, Maul abandoned his claim to the mantle of the Sith, focusing instead on building an empire of his own—the Crimson Dawn—by controlling the criminal underworld.
Like his master, Darth Sidious, better known as Sheev Palpatine, Maul's main weakness was his overconfidence. In Luke Skywalker's confrontation with Palpatine during The Return of the Jedi, he tells the Emperor that his overconfidence is his weakness.
In the end, it would certainly be a tough match no matter what -- but Yoda would likely come out on top. Anakin lost against Obi-Wan because of overconfidence and talking too much, and Yoda could have easily exploited this to his advantage.
The realistic answer seems to be, yes, Anakin in his prime could have defeated Luke Skywalker (in the current canon) if they battled one-on-one. The main reason behind this is Luke's lack of formal Jedi training, at least when it comes to combat.
