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Historically, units of measurement are often based on sizes of body parts. Probably the most obvious example would be a "foot" literally being based on the length of someone's (or everyone's, or anyone's) actual foot. Other examples include inch (length of a thumb, from tip to first joint), cubit (elbow to tip of middle finger), and span (from tip of thumb to tip of little finger, when spread as far as possible).

For most historical human populations that used these types of systems, this worked well enough for their day to day lives, despite the different sizes of body parts from one individual to the next. When that size variation was a problem, they could standardize, somewhat, by using the body part of a ruler/monarch/lord/etc. instead of just some random individual.

Enter the worldbuilding conundrum of populations that include both human and non-human races, some of which have distinctly differently sized body parts, or might even be missing a body part that another race has. The exact races present are not necessarily relevant to the question, but for illustrative purposes, I'll provide a couple quick examples. A lizard-human-hybrid type race would have a tail, that a human would not. A race of merfolk that are more fishlike than the standard trope might have webbing between their fingers or might not have fingers at all with fins or flippers in their place. Giants or dwarves/halfling-esque races might be mostly human shaped, but wildly out of proportion to humans in actual size.

In a world-building setting where at least 2, if not more, such races commonly and constantly interact, making body part lengths inappropriate as common units of measurement, what is the most likely basis for such a society to use as units of measurement instead?

You give your own answer!

When that size variation was a problem, they could standardize,
somewhat, by using the body part of a ruler/monarch/lord/etc. instead
of just some random individual.

There is some historic individual who has made a big impression on all of these involved races. Measurements reference body parts and famous exploits of this legendary explorer / philosopher / inventor / conqueror / lover.

This would be fun for a fiction because with each reference readers would wonder what all this individual got up to during his action-packed time. Maybe he even shows up late in the story.

Well, for one, it's unlikely that having multiple races would affect using body parts for measurement; humans come in varying shapes and sizes after all, and that's why we moved to more objective forms of measurement as our technology allowed for it.

At most, you'd wind up with a situation where they say things like "that's an orc-foot long" or "that weighs a dwarf-pound."

However, I imagine just waving away your question isn't a very satisfying answer, so let's try and figure out a solution through world-building.

Creating a metric system requires a significant amount of technology and understanding of mathematics and the sciences, and even modern SI units are poorly defined for certain dimensions (the speed of light is, by definition, 299792458 m/s, but the meter is defined as the distance traveled by light in 1/299,792,458th of a second, so you can see the problem, the kilogram was defined by a 200 year old reference standard until only a few years ago, and its replacement, a silicon sphere, isn't without controversy).

However, since it's a fantasy world, you can invoke a bit of magic, handwaving, or both.

You could have a tree that inexplicably always grows a branch of a roughly identical length, or have a race of people that always reach the same height upon reaching adulthood. You could have a magical spell that's been bestowed upon the world by God of Measures, that allows for proper measuring.

Another possibility is, depending on how savy your people are, or how well they can measure time, would be for them to measure the difference between two shadows of a stick at different times on a specific day. For example, your measure of length might be something called a "springbow" which is defined as the distance between a shadow cast by a stick at the 1st bell and 4th bell on the day of Spring Equinox. Assuming that the time of bells is well-defined, presumably through a well-regulated clock, one would expect this distance to be identical on the Spring Equinox every year.

As such, you could even have a "Festival Of Measures" where the townsfolk gather to observe the measuring of the springbow, and reference measuring sticks are provided to all in attendance for free.

Even this has some problems though, as for the springbow to be identical, it has to be measured at exactly the same time in exactly the same place and must neglect the effects of procession. Worse, you have to know exactly where to measure on your shadows, and measure from that point every time. Procession is easy to hand wave away; you could say it's small or nonexistent for your world. The place is also easy to define. You could use a marker in some important city in the kingdom, perhaps the Royal Courtyard or so. The last question, where to measure on the shadow, is much trickier, and becomes a question of "who measures the measurer?" However, this is a problem we, with all of our technology, deal with today as well. Even though we can 'define' what we mean mathematically by a kilometer or a volt, measuring it is not as easy, and we're constantly revising our system of measurement as our technology allows for higher precision measuring of it.

So in your fantasy world case, perhaps you could use last-year's official springbow reference stick as a kind of "ritual of renewal." This will limit, but not eliminate the error in measurement, but would still leave you with measurements that far exceeded the precision of the ancients, plus it provides for a good plot device that can lead to interesting storylines (what if someone steals the previous year's springbow, or it gets lost, for example).

The reason people used body parts was its something everyone could reference. Given that's not the case with multiple fantasy races, then they'd likely use some other common reference. Perhaps they'd measure lengths in bananas. Or Elephant tails. How much is that rope? Its 2 coppers per Narwhal tusk.

We have used non-human body parts for measurement before. Roman roads were just broad enough to allow for two horses to go side by side on them.

If anything, though, the usage of body parts from onespecies might make the other feel misrepresented. This may lead to tensions which will hasten the adoption of some sort of universal, scientific measure. The revolution from which this arises may be called in your world le metric tensor.

Different races world most likely still use their own body parts among themselves. When interacting with other races they would learn to change between units.

If the races have been in contact with each other for a long time, it would be easier to use measurement units taken from elsewhere in their nature, from animals and plants common enough that different races all know them. Trout length, birch leaf length, three ant lengths and so forth.

A system similar to a metric system is obviously the best, but that was not the question.

They could continue to use their own body parts, and simply be aware of conversions and other rules of thumb. If a human and a halfling are negotiating for some number of feet of cloth, they should establish which culture's "feet" are being used. (Or, maybe they don't and this is one way unscrupulous vendors try to swindle people in your universe.)

In pre-modern Earth it was common for well-established cultures to nonetheless have different units and different definitions of the same unit coexisting. Take the pous, the Classical Greek "foot", for instance. In Athens it was typically a hair under 300mm, but in Aegina it was about 10% larger.

Of course, this sort of untidiness leads to added difficulties in map-making, treaty drafting, trade, and taxation, so governments don't tend to like it very much. Among the accomplishments of most of your serious empire-builders, you'll find that they inevitably establish standard measures to be used throughout their empire. (I'm thinking here particularly of Qin Shi Huang, but there are many other examples.) Probably some of the "body" units in your story will fall out of favor as the cultures that promulgate them do the same, to be replaced by "standard" units based on those of the preeminent cultures.