I’m setting up my first limited company in the UK and just received my company number. I’ve also applied for VAT registration. Quick question—is the UK VAT number the same as the company number, or are they completely separate identifiers for different purposes?

No, they’re entirely different.
A VAT number is issued by Inland Revenue to any person or organisation, whether a multinational corporation or a sole trader, where their annual taxable turnover exceeds the predefined threshold, currently GBP85,000 for the 2021/22 tax year. It’s used for VAT purposes only.
A company number is only issued to businesses and organisations incorporated or registered at Companies House. These include companies limited by shares, companies limited by guarantee, public limited companies, limited liability partnerships, and limited partnerships. It’s used as a corporate identifier for a wide range of statutory requirements, ranging from submitting annual returns to filing corporation tax.
For more details, visit : https://businessglobalizer.com/blog/uk-vat-rates-essential-guide-to-correct-vat-calculations/