Let's say that my domain is website.com and that there is an existing websites.com.
Given that the websites.com (in plural) domain that I don't own has very high Domain Authority and the website.com domain that I own have very low Domain Authority, Is it going to be possible for my domain to rank up?
Do I risk getting penalized by search engines?
No, you do not get anything from it. If you consider that www.example.com and example.com are two domains according to Google, imagine two domains, from two hosters, with two different websites.
However, you should be carefull, because of Typosquating. If you have example.com, while exampleS.com exists, and you are in a similar industry or if people can't distinguish you from them, you might get some not-so-friendly message from their lawyer. If you do this on purpose to harm them, there might even be legal issues to this.
TLDR; Just pick a name you like/fits you.
The web is full of plural names (like googles.com) or the same names in other TLDs, If a popular website domain name is example.com then there is probably example.net registered amongst other alternatives (.org, .website, etc).
You won't get any more benefits from search engines than receiving visitors who wrongly typed the domain name (typos), you can't steal other websites authority, but you can have a slightly benefit from keywords being searched that match the domain. In this case the only real benefit is that users who instead of typing the popular example.com, they can wrongly type: exampleS.com and arrive to your web.
The only penalty that you might incur is if you have listed duplicate content on both the websites, else there should be no issue regarding the same.
