Then why Sega gave them two different names?
They didn't. They're both just Sonic. "Modern" Sonic and "Classic" Sonic refer to art styles and gameplay styles. They both feature the same character, named Sonic.
Also when you look at other characters such as Crash and Spyro you see that their "modern" incarnations/redesign are totally different characters than their original (like the Crash with the tattoo or Skylanders Spyro).
Skylanders Spyro is actually a different character from Classic Spyro, and they are both different characters from Legend of Spyro. However, the same doesn't apply to Crash. The writing and art style changed, sure, but canonically Titans and MoM take place in a series of events that includes the original trilogy. As such, "Classic" Crash and "Tattoo" Crash are the same character.
I am strictly against seeing Modern Characters in Classic games (even with a redesign) because they wouldn't fit nor respect the spirit of the classic era.
That's an incredibly limited mindset. Sonic, Tails, Knuckles, Metal, Amy, Vector, Espio, and Charmy all were able to bridge the gap from Classic to Modern. The opposite is every bit as possible. The same kind of design changes but in the opposite direction could very easily work to make any of the Modern cast fit in a Classic style environment. Think of it as if the Chaotix had first appeared in Heroes with their Knuckles Chaotix interpretations happening later. That kind of thing can work for any of the other characters as well.
That said, they would require such redesigns, which is why I brought up Shadow needing one to begin with.
I'm glad Whitehead didn't put modern elements into Mania.
Except like it or not, it does have them. In much the same way that SRB2 is able to take inspiration from more modern games and stylize it with a classic flair, Mania is the same sort of thing but in 2D. For example, the sections where you spin down funnels in Studiopolis are inspired by similar sections from Casino Paradise in Sonic Advance, and even use close to the same sound clips.