Because people think of something as "low" and "common", both of which it is, and then get themselves mixed up and say "lowest common denominator" which is not what they mean. (This vaguely similar to the way people say "steep learning curve" based on informal ideas of steep surfaces being difficult to climb, without thinking carefully of what a learning curve is and what steepness actually means for one.)
In short, the typical usage of "lowest common denominator" is a logical mistake. To speculate why it happens, my guess is that it's brought about by wanting (rightly) to use the terms "low" and "common", and the temptation of using (incorrectly) the technical-sounding term "lowest common denominator". That, and the fact that one has seen others use the phrase in that sense!
Using something that's actually logically right — like "highest common factor" — sounds positive rather than negative, so this is less used. (Although, in your HTML example, you could use something like "largest common subset of features", and be both right and understood.)
That's the answer, but explanation of the literal meaning…
To take the example of TV shows — because the citation given on Wiktionary is "Reality TV really is appealing to the lowest common denominator in audiences" — it is often said of low-quality populist shows that they are dumbed down, and at the "lowest common denominator", in order to cater to a large population. In this case, it is true that the quality may be low, and that the intention is to make something of a quality whose acceptability is common to the large population. But the resulting quality of the show is actually the greatest common factor! It is at the greatest level of quality that is still common (that's why it's low, because it needs to be common). If they were actually picking the lowest common quality, it would be even lower, or zero.
[Mathematical background: In mathematics, there are two related concepts: the greatest/highest common divisor/factor of two integers, say 15 and 24, is the greatest factor common to both, in this case 3. (Note that 3 is lower than 15 and 24.) The lowest common multiple of the integers is the smallest multiple common to both; in this case 120. (Note that 120 is greater than 15 and 24.) So the lowest common multiple is greater than the numbers, and the greatest common factor is lower than the numbers. This itself should be enough to suggest that if what you want to talk of is something low, you must use "greatest common…", not "lowest common…". But most people don't think so much.
Lowest common denominator is another term which is used when adding fractions: if you're adding the fractions 1/15 and 1/24, you convert them both to the lowest common denominator, which is the lowest common multiple of the denominators. In this case, the lowest common denominator is 120, and you write 1/15 as 8/120 and 1/24 as 5/120, so that you can add them:
1/15 + 1/24 = 8/120 + 5/120 = 13/120.
Even in this case, the lowest common denominator (120) is larger than the original denominators (15 and 24).]
The actual "lowest common factor" of any set of integers is 1, irrespective of what the numbers are, so it's not a very useful term. Similarly, the literal meaning of "lowest common denominator", when used in its usual context, refers to a quality that is always zero (or the minimum possible) irrespective of the population: the lowest common denominator among high-school graduates, the lowest common denominator among the whole population, and the lowest common denominator among people with PhDs would all be the same: abysmal.
Edit: Someone on Wikipedia has already explained this:
In common non-mathematical usage, the term "least common denominator" is often misused for the concept of the greatest common divisor. For example, a graphic toolkit which rendered features like lines and polygons into either Microsoft VML or standard SVG might choose to implement only the maximum set of graphic attributes common to both destination formats, which is an easy analogy to the concept of the greatest common divisor (The greatest common divisor of 12 and 18 is 6, which is the largest factor evenly dividing both numbers). If the systems being compared are very similar, then the common functionality can be a powerful subset (as the greatest common divisor of 375 and 250 is 125), while if the systems are very dissimilar the common capabilities might be very minimal (as the greatest common divisor of 270 and 98 is only 2). With additional systems (or numbers), the set of features common to all cannot grow and often shrinks (likewise for finding the greatest common divisor for a series of numbers).
This approach of making use of only the greatest subset of function common to all supported systems is often disparaged when the common feature set is sparse or weak (by analogy, having a small "greatest common divisor"). In this context colloquial usage has conflated the concept of "greatest common divisor" with the familiar sounding jargon of "least common denominator", which seems to emphasize smallness of overlap through the word "least", but actually refers to a different and inappropriate mathematical concept.
Edit 2: In case it helps, below is a vague picture illustrating what I mean. (Anyone please feel free to make a better image and replace this.)
When most people say "lowest common denominator", they mean the second horizontal line, near the bottom: it's very low, because it must be common to so many, but it's also not zero. If you take "lowest common denominator" in its mathematical sense of being an lcm (of some set of existing denominators), then it's actually higher than the elements of the set, and is the highest line. If you use "denominator" to mean just "trait", then the "lowest common denominator" is literally the lowest common trait, which is the very bottom-most line: it's always zero (or 1 in mathematics, if you're talking of factors), independent of your original population, and even lower than it needs to be.