Whatever happens anywhere its anandi's fault isnt it?
well, in Shiv's identity case she never pestered Shiv, or anyone for that matter to find the truth. She once didnt tell Shiv her convo with CM.
Also they both chanced upon the CCTV footage and got to know the truth.
And even if we consider that if anandi had brushed this off (which would have been morally wrong) and kept quite, then also the truth would have come out.
anandi just spoke to Shiv, not to Ira. Ira chanced upon the truth on her own. Infact it was Ira who first told everyone at home about nurse Sarla. After that when on Alok's manupulation Sarla called up Ira, Ira believed her. Anandi never pushed Ira or anyone in the family. Ira met Sarla when the latter was on death bed and truth came out in front of her.
So even if Anandi would have kept quite, still truth would have come out from Ira's end...
Now coming to why Anandi told this to Shiv, i think being his wife she needs to share everything with him, specially the truth related to his life. So she telling Shiv and going through this with him, being his pillar of strength is what she should have always done. This is how a husband wife relation should be.