Lotsa weird structures out there of course, and many are “slang” as opposed to “proper” names. Many are modern(ish) day creations by Politically Correct entities (such as the media, especially in the U.S.) that follow no logic that traditional speakers of English can perceive.
There’s a rule of thumb I learned ages ago to create the noun form used to indicate a resident of a specific country. If the country name ends in a vowel, you drop the vowel and add a simple suffix ending with “an” or “ian” to the name of the country (e.g., America to American, Canada to Canadian). A ‘sub rule’ if you will is to drop the last vowel or syllable completely: Germany to German, Turkey to Turk. If the name ends in a consonant then the most common suffixes are “ese” or “er”. The “er” ending seems to be commonly derived from Germanic language usage or syntax, and there’s a seemingly older English form to simply add “man” to the end of the country name. It also seems to me that there’s less "PC"ness to the way British English-speaking people use their language as opposed to American English-speaking people.
One thing to bear in mind is that the noun form we use in English is largely dependant upon our English-language-specific version of the country name. For example, the country known to us as “Germany” is known to its residents as “Deutschland”, and so the noun forms will be different depending on which name you use.
And last but not least, there’s always exceptions to the rule.
Random sampling of Countries + noun forms:
((to parse country name to noun form, try to use it in a sentence like this: “I am from [country] so I am a/an [noun form]”))
America → American
Canada → Canadian / Canuck
China → Chinese / Chinaman
Ghana → Ghanian
Mexico → Mexican
Puerto Rico → Puerto Rican
Deutschland → Deutschlander
Germany → German
Iceland → Icelander
Turkey → Turk
Iran → Iranian
Arabia → Arabian (or Arab depending on context)
England → Englander / Englishman
Viet Nam → Vietnamese
France → Frenchman
Netherlands → Dutchman
Nederlands → Nederlander
Wales → Welshman
Iraq → Iraqi
Scotland → Scot
Argentina → Argentinian
Bolivia → Bolivian
South Africa → South African
Sudan → Sudanese
Pakistan → Pakistani
Congo → Congolese
Egypt → Egyptian
I know a word ending in “ese” sounds like it should only be adjectival according to regular English grammar, but these forms are regularly used in **American **English as nouns whether or not they are “proper” under traditional English grammatical guidelines. I blame the media for the gross dumbing-down of America … those stupid language guidebooks (e.g., the AP Stylebook) that most if not all reporters use to vet their syntax are horrible – they deconstruct language to the lowest common denominator then use ‘newspeak’ logic to support their inventions of “simplified” spelling and grammar – for example they use ‘leaped’ [leept] instead of ‘leapt’ [lept] because they believe that the Average American™ is simply too stupid to understand that ‘leapt’ is past tense of the verb ‘to leap’ and similar such gross atrocities wrought upon irregular verbs. I’m tired of “hanged” instead of “hung” and “lighted” instead of “lit”. What’s next, “bringed” instead of “brought”, “leaded” [leeded] instead of “led”, “readed” instead of “read” ? Jeez, we already get “pleaded” instead of “pled” in every news report about court cases. Effin stupid reporters / media editors seriously tick me off.
But to more directly address the OP’s question, I don’t believe there is any appropriate English-language alternative to “Englishman” or “Frenchman” regardless of the subject’s gender.
