Grammar – ‘These Are the List of’ or ‘This Is the List of’?

The examples you give are plural nouns, but they have the characteristics of mass nouns in that they cannot take a numerical indicator (you don't say I have three savings).

The commonality is that these are nouns are formed from either verbs (participles) or adjectives by adding an "s". While nouns, they are often also used as adjectives, as in your example savings account.

I am not aware of a distinct name for these terms or a definitive list.

You can omit the word "like," and the phrase carries a slightly different meaning if you do, as you suspected in your question.

Per Merriam-Webster, "like" can mean

to be conscious of an inward impression, state of mind, or physical condition

In this definition, "like" functions as an intransitive verb, so it requires either an adjective or a prepositional phrase to describe how one feels, as you did when you wrote "I feel like I am part..."

However, feel can also be a transitive verb, as in this definition:

to have one's sensibilities markedly affected by

In such a case, the second part of your sentence could function as an object clause (for more on that, see here).

I feel [transitive verb] I'm part of the world [object clause].

Based on the definitions, the meaning could be considered different in each case. "I feel like I'm part of the world" means I am conscious of an inward impression that relates to being part of the world. "I feel I'm part of the world" means My sensibilities are deeply affected by the reality that I am part of the world. There is more of an implication here that you are part of the entity you're describing. The first use could imply that you are not part of something but you have a deep inner impression that you are. Consider:

I feel I'm part of the family. I feel like I'm part of the family.

The first example sounds like a confirmation that one is part of the family and that their sensibilities are deeply affected by that, while the second example leaves more room for the reader to infer that you are not part of the family, but feel as though you are.

These distinctions are minute, and generally speaking you're safe going either way. There is quite a bit of professional writing that uses the phrase the way you describe.

An example from The New York Times:

These are collages, and yet the artist feels he is part of the Cubist discipline.