To most speakers of modern English the verbs “to don” and “to doff” would be classed as old-fashioned language. The words evoke a sense of top hats and curtsy-ing and words like “m’lady”.
In fact, it is likely that many speakers would categorize “doff” and “don” as very formal language, while the semantic equivalents “to take off” and “to put on” would be informal or every day language.
But these verb forms are actually all the same! You see, in the Middle English period speakers had both the verbs “put on” and “do on” to describe the action of clothing oneself (or others, transitively).
By the process of coalescence, where sounds in words that are commonly said together fuse and then sometimes shift grammatically, the phrasal verb “to do on” became “to don”.
The same happened to the verb “do off” which became, by the mid 1700s, “doff”, progressing through a myriad of other forms along the way, including “daff”, “doft” , “daft” and “doff’t’st”.


These days the word “doff” primarily exists as a fancy way to remove one’s hat, with the phrasal verb “take off” having more every day duties and “remove” being more formal. But “doff” and “don”used to be far more widespread and casual verb forms, a pattern that is repeated often in English. Language that is deemed to be fancy now, is often just a remnant of ordinary historical languistic processes. And who knows, maybe in a hundred years speakers will be saying “pon” instead of “put on!”