Can the 2600 mA mobile power supply output 5V 1A?

It depends on the design of the circuit board. Big brands may be at most 1A, and many miscellaneous brands or small brands are basically between 500 Ma and 1000 Ma.

As far as the discharge capacity of lithium batteries is concerned, it is generally no problem in the range of 0.5C, and 2600MAH is a single battery, so it is no problem to do1.3a..