Can a battery be damaged by charging it with wrong amperage PSU?
You clearly have some misconceptions regarding how Li-Ion battery charging works.

The 5 V input does not charge the battery directly, there is should always be a charging circuit in between. If you would apply 5 V directly to a Li-Ion battery you damage it beyond repair and it might even catch fire.

There needs to be a charging circuit in between as Li-Ion cells must not be charged to more than about 4.2 V. So 5 V would be too much.

How much current is used to charge the battery depends on the battery charging circuit. The battery charging circuit decides how much current can be used.

It might decide that for example 1.5 A is needed. When you use a 5 V 2A power adapter like is needed for your device, all will be fine.

But if 1.5 A is needed but you have connected the 1 A adapter issues can occur. It can be that the 5 V voltage drops too much and charging is done at a lower current. It can be that the 1 A adapter is overloaded and shuts off. Note though that not many power adapters have an overload protection! The 1 A adapter might be overloaded, become too hot and be damaged and maybe catch fire.

What will happen when more than 1 A is needed from a 1 A adapter is anyone's guess as it depends on the design of the device you're charging (phone) and the design of the power adapter.

In general you should just use the recommended rating, so in your case 5V, 2A to be safe and not use an adapter with a lower rating.