Is a copied acknowledgements section considered plagiarism?
He recently realized that in the acknowledgements part the student has copied practically (word by word, line by line) the acknowledgements of another thesis available on the Internet by replacing the name of my colleague with the name of the supervisor of that thesis.


I'm slightly confused: I assume that in order to discover this, the advisor must have run the entire thesis through some plagiarism-detection software.  But the question indicates that this took place after the student's graduation.  I don't understand that: two reasonable times to run your student's thesis through plagiarism detection software are (i) at some point before you sign off on it and (ii) never.  Would one really do this after the fact out of idle curiosity, or did the advisor have some other reason to suspect that plagiarism took place?

Is line-by-line copying of someone else's acknowledgments a form of plagiarism?
Well, plagiarism has a sufficiently precise definition that I don't see how the answer can be "no".  The better question is whether this is viewed as unacceptable enough to do anything about.  

I think that part of that question depends upon the writing quality and originality of the purloined acknowledgments.  As @Andreas Blass points out, if you've seen 10 different thesis acknowledgments, you've seen 1000 more: such acknowledgements are formally part of the thesis but they are not part of the academic / intellectual content of the thesis.  So they are not vetted in any way, and there is no expectation of intellectual originality.  There is more than one way to phrase "Thanks so much to my parents, my friends and my thesis advisor", but the number of standard ways to phrase that is so much smaller than the number of theses that the pigeonhole principle applies to show that many people are writing the same sentences in their acknowledgments.  If the purloined acknowledgments contained many clever and original turns of phrase, then it looks worse: the fact that what you stole has nothing to do with the intellectual content of the thesis cannot excuse the fact that you are taking credit for someone else's distinctive writing.  That's a really icky thing to do in acknowledgments to your own thesis.

Unless the purloined acknowledgments contained several paragraphs of poetry or something similarly egregious, I would not consider a formal plagiarism inquiry.  This runs the risk of getting the degree rescinded, which to me is clearly too harsh.  But there is a real problem here:


  He added that he was hurt by the fact that the student has not tried to say his 'thank you' words by his own words.


Ça y est.  The student's former advisor has discovered that the student has cut corners and compromised some academic integrity just to get out of thanking him in his own words.  What a slap in the face!  I think that if something like this happened to me, I would have to bring it up with the student (though it would be painfully awkward) just because I would feel like I have to be honest with them about the change in my opinion of them resulting from this discovery.  Without some kind of sincere apology / explanation of this, the future of this professional relationship is at risk.  The student, alas, may need to be informed that he should look elsewhere for future recommendation letters and so forth.

Added: A lot of people seem to be suggesting that parts of this answer are an overreaction.  I tried to make clear that I don't find the premise of the question completely plausible or understandable, but after discussing that I did what I think is the appropriate thing on Q&amp;A sites like this one: I assumed the premise, in some form which made sense to me.  Yes, if the "word-by-word, line-by-line" copied acknowledgments constitute five lines thanking the student's family, friends and advisor, I see no problem at all -- and I  don't see it as a reason to run the text of the acknowledgment through an internet search.  Nor would such a search turn up a unique antecedent; it would turn up hundreds or thousands of cognate passages, as Andreas Blass pointed out. Thus to assume the premise I had to assume that the purloined text was much more substantial and distinctive. To be more specific, imagine the text was:


  I feel greatly honored to be able to add, in some small way, to a subject founded
  by so many [Field Xers] whose work I so deeply admire; among them are
  [My Advisor], [Person A], [Person B] and of course [Person Whose Name Appears in the Thesis Title], the richness of whose ideas seems undilutable by time.
  
  I thank [Professor C] and [Professor D] for suggestions directly relevant to the material appearing in this thesis. I thank [Fellow Student E] for helping me – quite a while ago now – with some results of [Somewhat Technical Theory] that appear in the appendix.
  
  I thank [Postdoc F] for helping me calculate a fistful of [Something] (I regret that none of these calculations appear in this thesis)
  and for generally being so free with his technical and [Field X-ical] insights.
  I am grateful to [Recent PhD G] for making his own as yet unpublished thesis
  work available to me.
  
  The graduate students in the [University Y Department of X] have been without
  exception intelligent and friendly, and it has been a pleasure to learn from them and with them over the years.
  
  Thanks to [Old Girlfriend H] for providing comfort in the sad days of [a few years ago]. Her kindness will never be forgotten.
  
  I am indebted to my thesis advisor [My Advisor] for more things than I can list
  here, but most recently for a careful reading of an early, ugly draft of a certain lengthy [Field X-ical] document.
  
  My father would have been proud of this thesis beyond all moderation. My
  mother has long made and continues to make me proud of her strength in the
  face of all the challenges life has to offer. I thank them both with all of my
  heart.


Imagine that the advisor read this the first time around and was happy to be thanked along with so many other people who made a difference in the student's life.  Then later, for whatever reason, he does an internet search on or including this text and discovers that it is lifted word-by-word, line-by-line from this 2003 PhD thesis.  The specificity of the stolen text would be downright creepy.  Is it really an overreaction if the advisor is upset by this??