According to the games' digital manual, the Motobug that Heavy Rider rides on is named Jimmy. This is in tribute to Polygon Jim, a member of the Sonic hacking community who died in a car crash in 2013.
Jim’s most prominent hack was Motobug the Badnik in Sonic the Hedgehog. In the hack the Motobug will extend its wheel out to jump, which the Heavy Rider’s Motobug also does.

The Studiopolis zone shares it's name with the studio where the English voice overs for the Sonic games, as well as Sonic Boom, have been recorded since 2010.

In Press Garden Zone Act 1, there's a hidden area that has word "HELLO" spelled backwards on the wall.

The sound effects heard during the boss fight against Heavy Shinobi in Press Garden Zone Act 2 are directly pulled from The Revenge of Shinobi.

According to Takashi Iizuka. Sonic Mania came out of the love from the original Sonic series. Christian Whitehead and Simon Thomley(the people behind the remastered Sonic ports on mobile devices) took part in the development of the game.

"Sonic Mania was born out of our fans love of the classic Sonic 2D platform games. This type of collaboration is a first for Sega and we hope everyone will be both surprised and delighted by this title. Sonic Mania has been a passion project for the entire team and we look forward to sharing more details about it later this year. Having the game actually playable at the event itself tonight was testament to the dedication of the team behind it.”

The gun-like launchers in Mirage Saloon Zone have an identical design to the guns wielded by the Eggrobos in Sonic & Knuckles.

In Mirage Saloon Zone various wanted posters can be found depicting Fang the Sniper (aka Nack the Weasel), Bean the Duck & Bark the Polar Bear, who also appear during the transition to Act 2 and in the boss battle against Heavy Magician.

In Hydrocity Zone (prior to Patch 1.3), if the player inputs from level select code from Sonic the Hedgehog 3 (Left,Left,Left,Right,Right,Right,Up,Up,Up) whist hanging on the hook levers (a ring sound plays if entered correctly), it cause a sound clip of Youtuber videogamedunkey to play whenever the players uses the dash panels.
The sound clip originates from his video Dunkey's Best of 2016, which Sonic the Hedgehog 3 was featured in.

The clapperboard in Studiopolis Zone has a timer that matches the time spent in the stage. And at the bottom it reads Sega, CW, HC and PWG, in reference to the developers Christian Whitehead, Headcannon and PagodaWest Games.

In Studiopolis Zone, the player can enter the SegaSonic Popcorn Shop machine, referencing the vending machine of the same name that served popcorn which also housed a minigame that could be played.

The TV the player pops out of after traveling through the satellite waves references the logo and design of the Sega Game Gear.

The van in Studiopolis Zone that can be entered to travel through satellite waves is branded with the words Hornet & High Class, referencing the Hornet Class car from Daytona USA.

In Studiopolis Zone there is a shop called Pink Bot, whose name and logo design are similar to the Pine Pot from the first stage of Streets of Rage.

Throughout Studiopolis Zone there are various signs for Club Spin referencing Club Sega, a chain of arcades in Japan, with Ages underneath simply being Sega spelt backwards.

After defeating the boss of Studiopolis Act 2, an error message pops up on the TV monitor in the background accompanied by a buzzing noise.
This was the same buzzing noise that occurred during the livestream of the Sonic 25th Anniversary Party event that Sonic Mania was first announced at, which became infamous for it's numerous technical issues.
Also the numbers 072216 displayed on the error message reference the date of the event, July 22nd 2016.

The words COPE and CPU can be found graffitied onto the subway trains in the background of Studiopolis Zone, referencing the same words that can be found in Spring Yard Zone from Sonic the Hedgehog.

The background of Green Hill Zone Act 2 was inspired by an early prototype version of Green Hill Zone from Sonic the Hedgehog.