It's to show humility, or to soften a claim, like this:
My new wife looked like an absolute bombshell at the party, if I do say so myself.
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But which is better, "do say" or "may say" ? Doesn't "do" sound too strong, like you are arbitrating something ?
When I sort of roll both choices around in my mouth, I don't detect any real difference in connotation. Â The word DO tends to hold a bit more stress than MAY, which gives the DO-sentence more audible strength. Â But I suspect this is just a natural phonetic thing, not a semantic thing. Â The difference in verb form is pretty much forgotten in standard phrases like this.
I use the do version. I can't recall ever hearing it with may. Either way I would treat this as a fixed phrase or idiom. Avoid parsing it out. It is neither asking permission nor posing any hypothetical. The elements have lost their individual grammatical and semantic meanings under the weight of the phrase as a whole.
Like Glenn, I have never heard "may" used in this phrase with "myself." Â What I do occasionally hear is, "If I may say" then they make their statement. Â These are times when someone is interjecting their opinion in a way that is a bit stronger than usual. Â This idiom seems to be asking permission to say this thing but no one ever waits for that permission to be given.
Glenn said
The elements have lost their individual grammatical and semantic meanings under the weight of the phrase as a whole.
It's a fixed thing, isn't it, sort of like with "What do we got?", the police at crime scene lingo.
The other one that Dick mentioned, "If I may say," is more like "If I may be so bold..."