A couple of bioengineers decide to play silly buggers with the exobiologists downstairs by cobbling together an artificial life form and claim it is a naturally-evolved alien animal
A DNA-based alien lifeform's DNA could theoretically be indistinguishable between "natural" or "artificial", because DNA is DNA (the likelihood of an alien lifeform to be based on the same DNA and DNA code as humans is small, but that's a moot point once the alien critter is delivered).
So, a DNA sequence that would fool the exobiologists could exist. Trivially, if the bioengineers had occurred by chance on the same DNA sequence of a natural alien critter, they would have the same sequence, which by construction cannot be distinguished from the identical sequence occurring in nature.
The question therefore is: what mistake did the bioengineers make?
There are several classes of possible mistakes.
The simplest to explain is the presence of "markers" of the DNA assembling technique used (for example, both CRISPR-Cas9 and CRISPR-Cf1 techniques rely on the presence of short sequences called protospacer adjacent motifs, and leave recognizable 'telltales'). If the DNA assembler used by the bioengineers is advanced enough that it can generate any DNA sequence whatsoever, these telltales will be missing.
Then, the DNA code is inefficient (or redundant, if you prefer), and presents "synonyms". Different DNA sequences yield the same meaning, and are equivalent, but they are not equally efficient, easy to synthesize or work with. So it is possible that the bioengineers' tool automatically employed codon optimization, and/or employed it differently from what a real living organism would have done. This kind of telltale is subtler and might escape a bioengineer.
Naturally evolved DNA, also, is a horrible, jumbled mess. Unless something occurs that actively selects against a given junk sequence remaining as a leftover in the DNA of an organism, that junk sequence will remain there, world without end. This "junk" is actually in some way functional (it acts as a scaffolding, of sorts), and is generated from existing genes from past generations, so it's not random and can be traced to specific sections of the DNA. Let us say that a DNA sequence spells the words "WHAT HATH GOD WROUGHT", you're likely to actually find "aaaqWHATxqaatathqHATHxsyndqGODxododwjehovroodwroqWROUGHTxjoe". Ancient versions of similar genes will be interspersed between the "active" genes introduced by 'letters' q and x.
Having no significant junk DNA, random junk DNA, or unrelated junk DNA would then be a very strong cause for suspicion. Engineered, "made" DNA is likely to be too efficient, too well laid out, or haphazard in the wrong way.
Finally, even if everyone is a bioengineer, still creating a working alien organism completely from scratch (and having it come alive! - which means thousands of the newly created genes actually working together successfully) would be too enormous an enterprise. It would be more difficult than, say, writing a word processor from scratch starting from the level of the assembly language.
The overwhelming chances are that the bioengineers would cobble together some library gene soup based on actual genes tweaked and redesigned by other people and found to be working. In the programming metaphor, they would employ a solid if basic language such as C, a working system library such as
glibc, a proven compiler. And probably recycle some existing word processor fragment. The result, if properly inspected, would reveal traces of all this.
Once the exobiologists find that the respiratory regulation gene AL3X7-P5 of their alien critter was published ten years earlier in an answer on Gene Overflow, the game would quickly be up.