My reading would be no.
5E is very simple in that it clearly states, you get what things say. You don't infer and you don't read between the lines. No Racial change, no sub race, no different ability modifiers.
You don't get to add or change your base HPs to the MM Doppelganger's, you don't get the special attacks, and you don't get it's immunities.
Your true race is what ever you picked at your character creation: Human, Elf, Dwarf, etc. and you would have all the HPs and stats that the race and your class combination gives you.
You get what ever is stated in the secret that is drawn/picked as a bonus flavor - very similar to background traits.
When using the doppelganger secret ability, any other form you take besides the one that you selected from initial character creation, is only for the look/feel. While being a human and taking the form of a Gnome you won't get its spell-like abilities or the Orc's fury ability - you retain all your Human abilities while taking the new form.
Here are some references:
About secrets from Perkins' interview:
“I think the most interesting part [of Rime of the Frostmaiden] is the character secrets,” Perkins said. “Characters can play it one of two ways. They can keep their secret close to their chest and not reveal it to the other players, fostering and breeding paranoia, or they can reveal it anytime they want to, and then wrestle with the consequences of it. That’s left entirely up to the players.”
The Icewind Dale includes a section called Appendix B: Character Secrets, and it has a list of seventeen potential secrets players can insert into their characters' backstories to help get them in the mood for a horror-themed DnD game. Like most campaigns, players are encouraged to develop their character's backstories, although many DMs will know that this can range from a vague one-sentence idea to several handwritten pages, depending on the experience level and style of the player.
About 5E basic rules:
Multiple articles discuss how not to create additional text that isn't there.
Two videos with Jeremy Crawford: