Is it plagiarism to copy an ethics statement on animal research? There are only so many ways you can say "This research was conducted in strict accordance with the guidelines of XX." I'd had the fear of plagiarism burned into me since high school (I think the rule then was 7 words or more verbatim were disallowed), but is it acceptable in science publishing to have substantial overlap with previously published ethics statements?

A slight variant is when you are describing a technique that has been concisely described by your lab dozens of times already, although this is slightly easier to think of variants than the above. But some amount of "copying" in these situations, whether inadvertently or on purpose, does not seem in the spirit of plagiarism. 

How do publishers/the scientific community see it?
Both plagiarism and self-plagiarism are about violations of the expectation of originality. In the main text of a publication, there is an expectation of originality, and thus these concerns apply.

In the "metadata" of a publication, like acknowledgements, statements on ethics and conflicts of interest, etc., the expectation of originality is much weaker, and the primary concern is accuracy.  Thus, "boilerplate" text providing little or no narrative content can be freely reused (e.g., "This represents the conclusions of the authors and not of the funding agency or US government.").  

In the unusual case where such a section does have study-specific narrative content (e.g., describing a novel approach to developing an ethics plan), then it would not be OK to copy it. Instead, you would want to discuss the relationship in the main text or methods section (which is a grey area for reuse, as sometimes it is "boilerplate" and other times narratively unique), and make a simple statement in your own words linking to that discussion in the ethics section.