I think the connection between those four nicknames/names is her identity. Cersei and Sandor called her "little dove" and "little bird" because of her innocence and her naivety. Probably, they both thought she was foolish.They use those nicknames more than her name Sansa.
Lady Lannister : Stannis called her Lady Lannister to convince Jon to join him and show him Sansa isn't really a Stark anymore.
Alayne Stone : this is her false identity in the Vale. The crown wants her dead, and she can't be Sansa Stark anymore.
During her time in KL, Sansa never became a Lannister. They tried to make her a Lannister, but in her heart she was still a Stark (for exemple, she felt safe only in the goodswood). In the Vale, she fights to stay Sansa Stark, but in the same time, she starts to think Sansa is dead and Alayne is another person.
Okay, I stop here with my long caption 😂