You would not actually be changing a PHP value--once the page is output to the browser, consider PHP gone--you can't interact with it because it's on the server. Instead, think of interacting with the document that's in the browser.
You will want to create some PHP code for handling the saving to the database on the Server side. You will POST the like / dislike value information to this server side script. If possible, I would use jQuery's AJAX helper to post data to the PHP page you just created.
Here is a great example of how you could accomplish this:
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About the technologies asked in this question
PHP (from the English Hypertext Preprocessor - hypertext preprocessor) is a scripting programming language for developing web applications. Supported by most hosting providers, it is one of the most popular tools for creating dynamic websites.
The PHP scripting language has gained wide popularity due to its processing speed, simplicity, cross-platform, functionality and distribution of source codes under its own license. https://www.php.net/
HTML (English "hyper text markup language" - hypertext markup language) is a special markup language that is used to create sites on the Internet.
Browsers understand html perfectly and can interpret it in an understandable way. In general, any page on the site is html-code, which the browser translates into a user-friendly form. By the way, the code of any page is available to everyone. https://www.w3.org/html/
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