Dude is a biological chimera:

A genetic chimerism or chimera (/kɪˈmɪərə/ or /kaɪˈmɪərə/, also chimaera (chimæra) is a single organism composed of cells with distinct genotypes. In animals, this means an individual derived from two or more zygotes, which can include possessing blood cells of different blood types, subtle variations in form (phenotype) and, if the zygotes were of differing sexes, then even the possession of both female and male sex organs[1] (...) Animal chimeras are produced by the merger of multiple fertilized eggs.

The fact that the first DNA sampling came out as a negative was just luck of the draw. Sherlock got a sample from one of the criminal's genotypes and compared it to the other by chance.

As for the fingerprints: it's usual for a crime scene to have the fingerprints of everyone who had been in there before the place got isolated, and the prostitute may have been visited by multiple clients (or it was a public place). The fact that they couldn't find a fingerprint match is a sign the detective did a sloppy job of comparing just the very first fingerprint they found to Jekyll's. That, or the suspect wore gloves.

He received a transplant. Maybe a face transplant! Someone should make a movie about that.

When organs are transplanted, they retain the original DNA of their donors.

This makes me think of Chimerism (mentioned in another answer) and Superfecundation

https://house.fandom.com/wiki/Chimerism

Chimerism is a very rare condition caused by the fusion of zygotes (fertilized eggs) into a single form during early cell duplication. It is extremely rare, with only forty or so known human cases.

https://en.wikipedia.org/wiki/Superfecundation

Superfecundation is the fertilization of two or more ova from the same cycle by sperm from separate acts of sexual intercourse

Heteropaternal superfecundation refers to the fertilization of two separate ova by two different fathers.

What's that mean?

You take different sperm from two fathers (mom had a fun night - or a bad night - and might explain the latent anger boiling beneath the surface), fertilize two eggs and then join them together early in the cell duplication stage.

Now you have one man with two sets of DNA.

Additional option

Since we are already talking about long shot odds on top of long shot odds... what if the eggs were implanted by invitro... what if the eggs got mixed and now you literally have two eggs and two sperms from 4 different people? Mix those together...

Result

What does the "potion" he takes do? It temporarily gives the other set of DNA control - physically and mentally.

Why doesn't the potion work on you? Because you aren't a Chimera or Superfecundated.

His DNA is always the same; it is the potion that filled his blood with molecules disturbing forensics exams.

• To be suitable to DNA exams, a hair must still have its bulb. Let's say they didn't find any.


• Fingerprints don't really yield DNA.

That leaves us with blood. Biological protocols are very subject to changes in pH, inhibitors and whatnot. It's sufficient that some molecule from the potion interacts with the standard kits and hinders the reactions necessary to identify its DNA.

Who knows, being a scientist he could have done it on purpose...

You're a Cephalopod in disguise!

Cephalopods have a unique ability to alter their own RNA! Because RNA regulates gene expression you can change how your traceable genetic material is seen from murder to murder. If your criminal didn't know about this ability its just a happy mistake, if he does and this doctor knows how to interpret this ability then he can actively suppress and release his genes in specific ways.

This does not change the issue with your finger prints. Thankfully Cephalopods have a solution for you! The way they alter their own coloring is by expanding and contracting small muscles across their body. Put a few of these bad boys in your fingers so you can change or smudge your own finger print while under questioning by the police.

Nanobots are rewriting his DNA when he transforms

It's hard to think of a plausible biological or chemical mechanism that can turn a normal-sized man into a "freakishly large" one and back again in such a short time frame. So how about this: Jekyll's potion either contains or activates a swarm of nanobots, which literally rebuild Jekyll's body before the detective's very eyes.

The 'bots suck in some matter from the surrounding air, pull some more from "less critical" organs (Hyde has no spleen and severely atrophied kidneys, since he never exists long enough to need them), and do some clever restructuring (Hyde's nano-latticed bones, while both longer and stronger than Jekyll's, actually weigh slightly less), and presto: the guy is a foot taller and has chimpanzee muscles (which are ~50% stronger than human muscles, though they sacrifice some endurance).

And while they're doing all that, they have to tweak his DNA too. Jekyll's DNA tells Jekyll's cells how to maintain Jekyll's body, not Hyde's; without the DNA changes, the whole biological system would start to fall apart the moment that the 'bots finished active construction.

Turning Hyde back into Jekyll is just the same process in reverse.

Where did the nanobots come from in the first place? Well, maybe Jekyll invented them, or maybe he salvaged them from a UFO crash site, or maybe they escaped from the lab upstairs and Jekyll doesn't even know he's using them...I'll leave it as an exercise for the writer.