Either way, in a just world, this company would defend the influencer against any trademark infringement claim and indemnify them against any compensation awarded.
In the real world, the opposite is usually the case, at least on paper.
Advertising placement agencies frequently hire influencers. These agencies use formal contracts with influencers that are one-sided in favor of the agency and the product maker. They generally hold the influencer liable for any liability arising from the influencer activity and require the influencer to indemnify and defend the agency and manufacturer from any legal action arising from the influencer actions.
Perhaps in practice, the manufacturer of the product would take action for infringement to protect its image and not irritate the influencer.
An influencer can be legally more secure if they just give a favorable review but don’t expressly advocate or provide a link to purchase the product.
I have not seen any case law on this approach, but, logically, the simple fact of pronouncing favorably on a product without advocating or facilitating a purchase could be interpreted as mere speech and not as advertising. No guarantees here. But advertisers usually want more, like an express call to buy the product and a link to where you can go.
Influencers should also review the Federal Trade Commission guidelines on deceptive marketing practices and influencer transparency.