Due to the fact that this is a lengthy question, I will try to be concise and to the point.
In short, yes, all of these main types coffee makers use a different type or size of filter. Sometimes each type can use a different filter. I will provide a non-exhaustive list of filter types for these coffee makers which you have listed. Although the list in non-exhaustive, it will provide the main types of filters for each device.
Understanding Permanent versus Paper filters
A permanent filter is often made out of metal and is, in most cases, a mesh (not true for espresso machines). They are not disposable and are made to be reused many times.
An impermanent filter is generally made out of paper and is disposable and is generally not meant to be reused.
Drip Coffee machines are one of the most common coffee machines for home use. The filter used will either be paper or permanent.
A paper filter for the drip coffee machine will either be flat-bottom or cone-shaped depending on the make and model of the coffee machine.
Cone shaped filter:
The permanent filter can take the same shapes, however it is generally a flat-bottom shape yet can be a hybrid between a cone shaped and a flat-bottom shaped filter. I can only describe this as an upside down triangular prism. Below is a picture of the aforementioned hybrid filter.
These filters are generally made out of paper and are usually a cone shape which is usually specific/unique to the type of single cup brewing device.
French (Push) Press
A french or push uses a permanent filter which is made out of a fine metal mesh. It is on the bottom of the plunger. The entire bottom of the filter has three main parts, shown below.
The Parts of a French Press Filter:
An espresso machine's filter is called a portafilter. It shaped like a small basket, made out of metal, and has small holes in the metal on the bottom. There are three sizes. The most common are the single and double shot filters and a rarer one is the triple shot filter. They increase in size to hold more ground coffee.
There are generally three types of portafilters in circulation with mainstream home espresso machines.
- Single wall. This is the most common in the industrial setting. It is contains only one wall on the bottom of the filter.
- Double wall. Think about this as a portafilter for the inexperienced and is most common for the home espresso machine. It has a secondary wall on the bottom of the portafilter with only a few holes in it. The presence of two walls helps to regulate extraction pressure and is therefore good for those who are not very good at tamping pressure and grind size.
- Pod filter. This is simply a specially designed filter that is the same shape as the others and comes with some home espresso machines. They are used to hold pre-made and pre-packaged espresso pods that one simply drops into the filter and can avoid many of the other steps of traditional espresso brewing such as tamping and grinding.
Portafilters (single wall) on left and baskets (which hold them and are upside down) on the right:
The filter on the left is a single shot and the filter on the right is a double shot.