Nope, they are only related through their Broadway fame. Like Lea Michele, Idina Menzel was offered Hollywood gigs after her great success on Broadway, originating iconic roles like Elphaba of Wicked and Maureen Johnson of Rent.
Interestingly, Wicked is one of Rachel's favorite musicals. I wonder how she feels about Spring Awakening, which launched Lea's career...
Below is a photo. On top is Idina Menzel playing Maureen and Elphaba respectively. On bottom is Lea Michele with Jonathan Groff in Spring Awakening.
The actress playing Rachel's mom is Idina Menzel. No, she and Lea Michele are not related. Before Idina was even cast Glee fans saw their resemblance and petitioned for Idina to be cast as Rachel's mom. Then Idina was cast as Shelby Corcoran (the Vocal Adrenaline coach) the petitioning continued and either Ryan Murphy listened or he thought Idina's character should be Rachel's mom too. Her casting was phenomenal! And I agree, their resemblance is amazing. :)
On Wikipedia, it said they had already cast Idina Menzel as the coach of Vocal Adrenaline, but then fans thought that she looked like Lea Michele so much and they petitioned or something like that for Idina to be Rachel's biological mum.
Unfortunately no Rachel(Lea Michele) and her mom Shelby Corcoran(Idina Menzel) are not related in real life. Idina was very well cast as her mom for the resemblance as indeed very good and their voices are amazing together. I wondered this myself for a while then I googled rachel and her mom and found info on the glee cast and who plays who.
They are "related" per their love of Barbra Streisand. When their idol was recognized at the Kennedy Center Honors, Idina Menzel sang "Don't Rain on My Parade," changing the Mr. Arnstein line to "Hey there Miss Streisand, I am your biggest fan!" I was stunned by the resemblance, particularly given the circumstances!