That's not legally correct in the US, EU, or the UK. Private ownership gives Amazon a lot of discretion over its own site design, messaging and whatnot, but not unlimited freedom to do or say whatever they please.

In the US major firms do not get a free pass simply because they own the platform and the idea that a website constitute "private property" doesn't work as a defence to anticompetitive conduct or to display a political banner expressing support for a political party of candidate without triggering additional rules / limits.

In the EU this is even less the case, as it effectively treats some platform conduct as capable of creating societal/systemic risks and thus needs to be kept in check. Whether is happens like that all the time in reality is subject of another discussion, I think; the point is that the mechanisms exist.

Political spending/advertising is a regulated activity that goes beyond rules that apply to private property. In the UK, for example, spending, donation, reporting etc. if the activity is intended to influence voters, falls under specific regulations: https://www.electoralcommission.org.uk/our-guidance/campaign...