To answer the title question, "ache" is definitely not an open syllable.
However, the letter "a" used to spell the word "ache" could be considered to be in an orthographic "open syllable", depending on how you define that term in the context of analyzing English orthography.
"Open syllable" means different things in phonology and phonics
As far as I can tell, that site is using "open syllable" in a "phonics" sense, to refer to a certain pattern that exists in writing, not necessarily in actual spoken English. Even in the context of phonics, it seems somewhat uncommon to describe a word like "ache" as having an open syllable: most other accounts I can find classify words spelled with "silent e" as having a special, separate type of syllable. But there is some logic to classifying the vowel in a word like this as being in an open syllable.
The written representation of the word "ache" doesn't end in a "consonant letter", it ends in "silent e" which can be viewed as a "vowel letter", so it can be graphically divided as "a-che", putting the letter "a" in an "open" orthographic syllable for the purposes of phonics even though the word only constitutes one spoken syllable. This is a way of explaining why the letter "a" in this word corresponds to the "long a" sound (IPA /eɪ/) rather than "short a" sound (IPA /æ/).
It's useful to be able to refer to this concept when teaching English spelling. It's unfortunate I guess that this doesn't correspond to the usual phonological concept of an open syllable, but then again, actual phonological syllabification is pretty confusing anyway in English (as far as I know, there is no definite consensus on how to syllabify words like "lettuce").
I have heard of some synchronic phonological theories of English that feature a similar idea (a postulate that word-final consonants in English are, or can be, "extra-metrical", and that this explains certain violations of usual phonotactic patterns in this context) but I don't understand this idea well enough to explain it, and I don't think it's directly related.
From a historical perspective, many (but by no means all) orthographically "open" syllables were indisputably phonologically open in Middle English: e.g. the word "name" was at one point pronounced /ˈnamə/, which would be syllabified as /ˈna.mə/, and a sound change that applied in Middle English to certain vowels in certain types of open syllables (called "open-syllable lengthening") caused the "a" to lengthen, resulting in /ˈnaː.mə/. That's why the word "name" is pronounced with a "long a" sound (IPA /eɪ/) today. This sound change is the origin of the spelling pattern that associates word-final letter "e" with a "long" vowel earlier in the word (since in early Middle English, word-final schwa was written with the letter "e"); however, word-final schwas were lost due to another Middle-English sound change in Middle English, and as a result the "silent e" spelling pattern came to be extended to many other words that had long vowels for unrelated reasons, such as "wife", from Old English wīf. (I believe I've read that during the time period when word-final schwas were being lost, there also seems to have been some amount of actual extension of phonetic schwas, not just purely written "e"s, to words that etymologically shouldn't have had them, as a result of the growing confusion between these two classes of words.)
To sum it up, perhaps you could say that at some layer of abstraction, this feature of English orthography is related to the phonological concept of open syllables. But it's not normally considered the same. If a connection doesn't make sense to you, you can just consider these to be accidentally homophonous terms that refer to completely different things.
Variation in the use of the term "open syllable" in phonics
As mentioned in the first paragraph, the source you found might be considered unusually general or sloppy in its use of "open syllable" to describe the "a" in "ache". A number of other phonics sources I found use more extensive classifications of syllable types that treat syllables ending in "silent e" separately from orthographically open syllables; e.g. "Six Syllable Types" from Reading Rockets uses the categories of closed, VCE, open, "vowel team", "vowel-r" and "consonant-le" syllables. However, the basic similarity between the behavior of vowel letters in VCE contexts and in open-syllable contexts is noted by Jakub Marian ("Pronunciation of vowels in open and closed syllables in English"), who writes
The letter “e” plays a somewhat special role in English. Apart from its regular use [to represent a vowel sound], it is often used to complete a syllable in order to make the previous syllable open. For example, in the word “take”, the final “e” remains silent, but is used to make the word optically two-syllabic (i.e. ta-ke), thus changing the sound of a from /æ/ to /eɪ/.