Section 194P

IF THE OTHER BRANCH IS ALSO CONNECTED UNDER CORE BANKING SYSTEM, IT SHOULD ACCEPT THE APPLICATION.

Yes. He can claim since the bank is same but barnch is different

Extract from section 194P for definition of specified person is as below.

 "specified senior citizen" means an individual, being a resident in India—

  (i)  who is of the age of seventy-five years or more at any time during the previous year;

 (ii)  who is having income of the nature of pension and no other income except the income of the nature of interest received or receivable from any account maintained by such individual in the same specified bank in which he is receiving his pension income; and

 (iii)  has furnished a declaration to the specified bank containing such particulars, in such form and verified in such manner, as may be prescribed.]

Therefore as per clause(ii) above he is eligible to claim the benefit u/s 194P