Why does Microsoft Word mark contractions as a grammatical error?

Maybe it is default checked in 2016, while it was not in previous versions. You can change it in

• File - options - proofing
• next to "writing style" click "settings"
• in "style" group uncheck "contractions"

As far as I know contractions are still considered proper English.

They are - but not in formal English.

Note that this a Grammar setting not a Spelling setting.

Please take a look at the Grammar and Writing Style options, in particular the section called "Formal Language":

• Contractions Targets contractions (e.g., let's, we've, can't) which should be avoided in formal writing, such as in legal documents. Example: The animal won't be authorized to be out of the bag during the flight. Won't will be corrected to will not.

• Informal Language Targets informal words and phrases which are more appropriate for familiar, conversational settings. Please consider using more formal language. Example: Our atmosphere includes comfy massage chairs. Here comfy is corrected to comfortable.

• Slang Targets regional expressions or slang terms which may not be understood by a general audience, and should therefore be avoided in formal writing. Consider using more standard expressions. Example: My cat barfed all over my homework last night. Barfed is corrected to vomited.

You need to uncheck "Contractions" in the "Grammar" settings, first selecting "Grammar & more" in the "Writing style" popdown:

Instructions:

• Click the File tab, and then click Options.

• Click Proofing.

• Under When correcting spelling and grammar in Word, click Settings.

• The Writing style menu has two options Grammar and Grammar & more. You can choose either option depending on which settings you want apply to your document.

Tip: By default the editor proofing options are set to Grammar & more and have Wordiness and Nominalizations style options selected.

• Choose Grammar & more option from the drop down if you'd like to have suggestions for style

• Scroll down to see all of the options available, and select or clear any rules that you want the grammar checker to flag or ignore. Any changes that you make to these settings apply to all the documents or items that you edit, not just the current document you are working in.

Source Select grammar and writing style options in Office 2016

Image source Checking style in proofing in Word 2016