Some similarities:
Both of them think the world revolves around them. They think any issue, news will affect them the most.
Both have around 5 gms of maturity.
Both are arrogant and don't know how to talk to elders.
Both take hasty decisions based on their emotions. Gauri going 'I don't care about auspicious day. I want to get married now, that's it' and Sanchi going 'Shiv can get engaged to anybody he wants. I am not attending the engagement': both scenes look similar.
The biggest similarity: Both hate Anandi.
What other similarities can you think of?