As with most IRL ethnicities, not only are people often considered "indistinguishable" by outgroups (as in "all chinese look alike" or "all blacks look alike" —often uttered by people with few or no friends of these ethnicities)
There also is a tendency to exaggerate so-called "identifiers" and elevate them to "racial" —even racist— stereotypes.
The most believable, less weird, interpretation, is the idea that dwarf women have a tendency to grow a "fuzz" which may or may not be called a "beard" by outgroups, comparable to the "sideburns" or "moustaches" noticeable in some black or southern or middle-eastern (south-american/south-european) ethnicities.
Remember how Frida Kahlo vindicated this "moustache"? Look up pictures of her, and you will see it wasn't that dark or all that "shocking". Hence the fuzz, as in above images, that may or may not be shaved, trimmed, bleached or hidden, depending upon the status, traditions and practices in said dwarven subculture or family.
This interpretation takes into account that Tolkien's books are considered being written by ELves or Humans… both outgroups to the dwarves.
The idea of female dwarves being indistinguishible from male dwarves echoes the ideas of Viking societies in which some women (even lesbian women) were known to join in the fighting/pillaging, wearing "male" armor and weapons, but also echoes to me how until recently any woman who spoke up, or took part in public life, and used the language of male society, were considered "boorish" and "unladylike". Classical Japanese culture is an example wherein women are demanded to remain "ladylike", but men are demanded to remain "boorish": they have distinct speech/language and behaviour… Tolkien's rather male-oriented society (kings, not queens) —even in Jackson's more feminist interpretation— may believably interpret a less "gendered" society (at least in language and movement) as being "one-gendered".
Another idea may be that most dwarves are (on the contrary) extremely jealous of their women appearing in public (echoing many human cultures IRL), which may render concepts of the "female dwarf" as something only dwarves could have information on. Gimli's remarks that they get mistaken for male dwarves, can echo both possibilities of a less-gendered "egalitarian" society, or a highly gendered society in which women are "hidden"/"disguised" when leaving the dwarves' abodes.
Aragorn's "It's the beards" seems the kind of joky, playful remark, resembling a reference to the "moustache" of latinx women. Racist stereotype, yes, but probably okay between fighting buddies travelling together, sharing the same hardships.