I don't think the plural would be used to intensify the meaning, and I do sense like (pob14 and Packard) that "man of his word" is the preferred alternative. But I also think many people use "a man of his words" to indicate the same meaning as "a man of his word."
If a man of his word means he keeps his promises, can I say I'm a man of my word to mean I am a person who always keep my promises. The question may sound a little silly, if you want to laugh, feel free to laugh, I've noticed that it's regarded as a set phrase, so I'm not really sure if the subject can be changed to I, or she.
He's a man of his word.
She's a woman of her word.
They are men of their word, we are men of our word, etc. I wouldn't use the plural "words", even with a plural subject, as I prefer to stick to the set phrase.
I don't recall hearing the phrase used with the plural words. Like veli, I prefer to stick to the set phrase. There are also other phrases with the singular word, eg 'my word is my bond', 'to give somebody your word', 'to keep your word', etc. They don't work with words.