Yes that’s true.
It’s a specious argument to say government finance is rather like household spending. It oversimplifies matters, and plays on poorer people’s fear of debt and its consequences. That’s a handy political line if you want to justify public spending cuts (which impact on the poorer sections of society), as well as overseas aid to starving countries like Yemen, etc.
It’s similarly misleading to talk about future generations repaying the debts of today, as if they’ll be sooooo unhappy! This is an old political trope. Debt to the US in 1945 was only paid off in 2006!
However, anyone paying tax in those years was presumably comfortable about the need to incur and pay off a debt which was used to make people safe and rebuild the country (including the NHS), just as future generations should understand the need to finance rebuilding today, while bearing in mind the calamitous political failings which brought this about.
Truth is, to live in a decent society we have to pay taxes at pretty much any and all stages of our lives, and that includes paying for the mistakes, tragedies and follies of the past.