Rising from the Last War text: doesn’t say
The text of the ability from Rising from the Last War quite simply doesn’t give us enough information to say one way or the other on this question. It lists a number of ways in which you definitely can change your appearance, but the use of the word “including” to introduce the list indicates it is not exhaustive: you can change your appearance in these ways, but you can also change your appearance in other ways not listed. The absence of evidence here cannot be construed as evidence of absence—we certainly can’t say from this text that you can, but we equally cannot say that you can’t. There just isn’t enough information to say.
Wayfinder’s Guide to Eberron text: doesn’t say, but might imply “no”
Wayfinder’s Guide to Eberron was an Unearthed-Arcana-tier publication that previewed some of the content in Rising from the Last War, and it includes a version of the changelings. The text of this ability is lengthier there, and includes
You decide what you look like, including [...] any other distinguishing feature.
Which sounds kind of hopeful—it’s far broader than anything on the list in the Rising from the Last War version of the ability. However, really, both Rising from the Last War and Wayfinder’s Guide to Eberron indicate a non-exhaustive list (both introduce it with “including”), Wayfinder’s Guide to Eberron is just a little more explicit that it is non-exhaustive.
Further on, however, it says
Your clothing and other equipment don’t change [...] requiring you to keep a few extra outfits on hand to make the most compelling disguise possible.
Now, this doesn’t strictly rule out faking clothes with this ability—after all, the fakes are never going to be as good as the real thing, and you wouldn’t be able to, for example, take off a fake coat or hat. But it certainly goes farther than Rising from the Last War in suggesting that clothes aren’t happening.
Eberron Player’s Guide text: definitely cannot
Going even further afield than Unearthed-Arcana-tier content, Eberron Campaign Guide was the 4e book describing Eberron, and it described changelings for that edition.
Effect: You alter your physical form to take on the appearance of any Medium humanoid. You retain your statistics in your new form, and your clothing, armor, and possessions do not change. The new form lasts until you change form again.
In this case, you could only fake clothing if the Medium humanoid you were trying to take the appearance of had fake clothing as part of its body—which is probably none of them. So in 4e, changelings definitely could not do this.
Eberron Campaign Setting text: probably not
The original description of Eberron and its changelings was Eberron Campaign Setting, for D&D 3.5e. There, we have
Minor Change Shape (Su): Changelings have the supernatural ability to alter their appearance as though using a disguise self spell that affects their bodies but not their possessions. This ability is not an illusory effect, but a minor physical alteration of a changeling’s facial features, skin color and texture, and size, within the limits described for the spell.
And then for disguise self we have
You make yourself—including clothing, armor, weapons, and equipment—look different. You can seem 1 foot shorter or taller, thin, fat, or in between. You cannot change your body type. Otherwise, the extent of the apparent change is up to you. You could add or obscure a minor feature or look like an entirely different person.
Here, disguise self can explicitly disguise your gear, but minor change shape nixes that by restricting it to “their bodies but not their possessions.” Minor change shape also includes a list of things you can change, but it is not introduced with “including”—which may indicate it is an exhaustive list. And it definitely does not include faking clothes, unless perhaps they are skin-tight and you can achieve that effect via skin color and texture. Of course, treating this list as exhaustive is really problematic, because it prevents all kinds of things that changelings are described as doing—by this list, you wouldn’t be able to bulk up, slim down, and so on, all of which we know changelings can do, both from other editions and from various descriptions of changelings within 3.5e.
Conclusion: Probably safest to assume “no.”
The official rules for 5e changelings, as found in Rising from the Last War, neither prevent nor allow this behavior, but the rules from earlier publications all suggest—if not outright state—that this is beyond their power. There’s certainly room to ask your DM for it—and if they’re willing, then awesome, because it’s a cool idea and I strongly doubt it’s going to be a problem—but there’s a lot more room for the DM to say no.
The one exception to that might be skin-tight clothing, which seem to be within the capabilities of all versions of the changeling race aside from 4e’s (and 4e was weird in a whole lot of ways).