No. This is bunk. A cleric of Eldath who has the nature domain will worship in a different way than a cleric of Eldath with the Life domain, and likely be in different sects. Hell the cleric class even says you are wrong, "As a cleric, you choose one aspect of your deity's portfolio to emphasize, and you are granted powers related to that domain." You emphasize the domain(roleplay) and are additionally granted powers related to it. The Acolyte only has the RP portion.
Remember that thing about highlighting the wrong parts of your quotes? You might have missed the part that says "
and you are granted powers related to that domain." Clerics choose an aspect to emphasize, as a domain that they follow, because it grants them powers.
An acolyte doesn't choose a domain. You can have an Acolyte of Lathander as a War God who leads his people into battle. They pick aspects, sects, how to worship, but the "Domain" is a purely mechanical aspect that Clerics utilize as their subclass. Again, the part you are quoting and relying on is talking about domains for clerics.
Actually, I just went to find the quote and realized you aren't even quoting the Acolyte Background. The Acolyte NEVER uses the word domain. You were quoting Appendix B. Let's look at the full paragraph here, not just that sentence.
"Your DM determines which gods, if any, are worshiped in his or her campaign. From among the gods available, you can choose a single deity for your character to serve, worship, or pay lip service to. Or you can pick a few that your character prays to most often. Or just make a mental note of the gods who are revered in your DM’s campaign so you can invoke their names when appropriate. If you’re playing a cleric or a character with the Acolyte background, decide which god your deity serves or served, and consider the deity’s suggested domains when selecting your character’s domain."
If you are playing a cleric or a character with the background, decide which god your deity serves or served.
Consider the deity's suggested domains
when selecting your character's domain.
The acolyte doesn't select a domain. Nowhere in the Acolyte background does it call on you to select a Domain. The only time your character is officially called on to select a domain is as a cleric, because your domain is your subclass as a cleric. You are taking the idea of religious sects, then tying it to domains, which you don't need to do. You can have two sects worshiping the same god in the same "domain" but just in different ways. The concepts aren't linked. And you are doing this solely so you can say that the Death Domain, which is the ONLY domain listed for Iuz, is listed for Iuz because the non-spellcasting acolyte needs it to know how to properly pick which way they worship Iuz.