Should a page have only one H1?
One, in almost every case. The H1 is the page announcing its own subject, and an announcement made twice with different words is not an announcement — it is ambiguity that a machine has to resolve by guessing.
What it actually is
Headings are not font sizes, although that is how most sites end up using them. They are structure: the H1 is the title of the document, and everything under it is subordinate. When a page has several, it is claiming to be several documents stapled together, and anything reading it has to decide which one it really is.
Why it matters
The H1 is one of the few signals that both machines and assistive technology depend on in the same way. Screen reader users navigate by heading, and answer engines pulling a citation need to know what the page is called before they can decide it answers anything. A page whose subject is unclear tends not to be chosen for either purpose, and the loss shows up as absence rather than as an error.
What good looks like
Exactly one H1 near the top of the page, stating that page's topic in plain language, and closely related to the title without being a word-for-word copy of it. Everything else that looks like a heading is marked up at the level that reflects its place in the structure rather than the size it happens to be.
Being honest about it
Search engines have said for years that several H1 elements will not get you penalised, and that is true — this is not a rule that carries a punishment. It is a clarity signal rather than a compliance one. Where it genuinely matters is on pages whose subject is not otherwise obvious, and on sites where the same template produces the same muddle across hundreds of pages at once.
Where most sites go wrong
The most common failure is the logo. A site puts its name in an H1 in the header, every page inherits it, and every page therefore announces itself as the company name rather than as its own subject. The other frequent one runs the opposite way: a page with no H1 at all, because the designer wanted smaller text at the top and reached for a lower heading level to get it.