Can one bank be the correspondent bank of itself?
Yes it is possible, I worked for a big bank subsidiary in a different country. We had a nostro account with the parent bank, and with its other subsidiaries. I cannot comment on the benefits/limitations of this though.

EDIT:
I remember we did not pay fee for the nostro accounts held with the parent and subsidiary banks. Moreover, these accounts were necessary as we had to deal with transactions involving various currencies. If let's say a client wanted to deposit in EUR, the bank should have a EUR nostro otherwise, the bank would not accept euro deposits.