The main reason why corporate personhood exists is to limit the liability of owners. If I run a company and a customer slips and falls, they can sue the company, but they can't sue me or my shareholders. Without any form of personhood there could be no limited liability. The customer would be suing me and the shareholders directly.
I don't think it would change much for giant corporations but that would be terrible for small businesses. I have a friend who makes his own stuff and sells it to people. He doesn't make much, a few thousand a year. He incorporated to prevent losing his house from a customer suing him.
I once worked for an Unlimited Liability Corporation (ULC). It's a corporation where the owners and shareholders can be sued directly for company actions. They took on that higher risk because the tax breaks that come with a ULC were worth it I guess. So it's not like giant corporations wouldn't exist if they weren't considered people, but it probably would hurt entrepreneurship.
The main issue Americans have with corporate personhood is the "freedom of speech" thing the US Supreme Court ruled on in Citizens United v. FCC. The ruling basically said corporations can't be prevented from giving their money to political causes because its a violation of the corporation's freedom of speech. That's specifically a US ruling. Other countries don't grant rights and freedoms to corporations.