What is a modal, really?
A miserable pile of words that together form a verb phrase.
Seriously, though, first look at how English can be considered to have "verb forms" that consist of two or more words, containing be and have as a "helping" or auxillary verb. This occurs with perfect tenses, passive forms, and progressive/continuous forms.
I have walked to the store.
I am walking to the store.
The paint was applied on the car by the painter.
Modals kinda-sorta follow the same principle and are additional "option" words that further modify the verb and make it conditional and/or add other shades of meaning relating to predicting the future and/or expressing indirect requests/commands.
From Wikipedia, here are the modals and words that can sometimes be modals:
The principal English modal verbs are can, could, may, might, must, shall, should, will and would. Certain other verbs are sometimes, but not always, classed as modals; these include ought, had better, and (in certain uses) dare and need. Verbs which share some but not all of the characteristics of the principal modals are sometimes called "semimodals".
To turn to walk, for example, into present perfect form, you would say I have walked. Have changes according to subject, not walk anymore. E.g. I have walked, he has walked, he/she/it has walked.
The modals operate the same way but don't change form according to subject. E.g. I may walk, he may walk, he/she/it may walk. So it's part of a verb, but modals never occur without a verb. If they do, part of the verb is being conversationally elided. E.g. "I knew he would." = "I knew he would [do that]."
Modals + to have and to be in the perfect/progressive forms exist. E.g. I may have walked, she was walking but not in the passive form unless you use have - e.g. The paint was applied by the painter -> The paint may have been applied by the painter.