Can a battery (specifically a lithium-ion type that you would get in a small portable electronic device) be damaged by charging it with a 5V 1A PSU, where the manufacture specifies as 5V 2A PSU?
Similarly can a charging a device that requires as 5V 1A PSU with a 5V 2A PSU damage the battery?
If the battery was not damaged in either of the above examples, could other electronics in the device be?
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.
First and most important, it is not safe to charge a battery with higher-than-specified current. Doing so risks damaging the battery (at best) and causing a fire or explosion (at worst).
Fortunately, what you are asking about is not the charging current, but the current available from the power supply that supports the charger. The actual battery charger is inside the "small portable electronic device" and the charger itself will limit the charging current to a safe level. The charger will draw only as much current as it needs from your PSU. The PSU you select should provide the same voltage that the electronic device expects and at least as much current as it needs.
The charging device should always match the cell's specifications, even though that cell has a certain tolerance (the electronics that comes within). Try to read here, there is a good resource: https://batteryuniversity.com/learn/article/charging_lithium_ion_batteries
Not properly used the Li ion cell could even explode.
