No. The various Fediverse server applications are too different in how they work.

First of all, it isn’t all just about object types. The architecture of various Fediverse server applications is vastly different, including how they handle objects, and how they distribute them.

For example, on Mastodon, a thread is just a loose string of posts and more posts which, technically, are identical in properties. Mastodon doesn’t know conversations, and Mastodon doesn’t know groups. You receive the posts from those whom you follow plus, by default, the posts that mention you.

Friendica does know conversations, and it knows groups because it has them implemented. On Friendica, a thread is one (1) post plus comments, just like on Facebook or on blogs. You receive the posts from those whom you’re connected with, but not their comments on other people’s posts. Plus, you receive all comments on posts from those whom you’re connected with. Receiving posts from those whom you’ve mentioned is optional but off by default AFAIK.

Forte is like Friendica, but with nomadic identity. That obviously isn’t a client thing.

Hubzilla and (streams) are like Forte, but with wholly different protocols that were made for nomadic identity in the first place and with ActivityPub as an optional extra.

Lemmy, Mbin and PieFed are all about conversations and groups. You literally can’t follow Lemmy users (something that Mastodon users will never understand), you can only follow Lemmy communities (something that’s totally alien to many Mastodon users).

There are many more differences.

Mastodon’s HTML sanitiser that rips out most text formatting is on the server side AFAIK. If you make Mastodon the gold standard, say buh-bye to numbered lists, horizontal lines, tables etc. (And I’m not kidding, there are places in the Fediverse that support these. In posts.)

Character limits are server-side. Since the huge majority of Fediverse users and many Fediverse devs think the Fediverse was made as a Twitter replacement, they also think that there has to be an arbitrary character limit, otherwise it wouldn’t be microblogging, right? Welll, then Friendica, Hubzilla, (streams) and Nomad can wave good-bye to their unlimited character counts and 100,000±character posts.

Filters are server-side. And they work vastly differently on different Fediverse server apps. Some import filtered content and then delete it. Others reject it.

Permissions are server-side. Permissions are absolutely essential and integral parts of Hubzilla, (streams) and Forte, but the entire rest of the Fediverse doesn’t even know they exist. Of course, it’d be great if everything down to mastodon.social implemented the (streams)/Forte permissions system, but it’d completely overwhelm those who came to mastodon.social in search of Twitter without Musk.

Another feature that Friendica and Hubzilla could kiss good-bye if there was only one unified server backend are multiple profiles per account. Speaking of which, it’s farewell to multiple channels (identities, like accounts everywhere else) on one account/login for Hubzilla, (streams) and Forte. Unless everything else is willing to implement both.

Lastly, Hubzilla has absolute literal shit-tons of features on top of even Friendica. Both have built-in file spaces, but Hubzilla has one with WebDAV connectivity (as do (streams) and Forte). Both have federated event calendars, but Hubzilla also uses it as a frontend for its built-in CalDAV calendar server (which is headless on (streams) and Forte). Hubzilla, (streams) and Forte have an optional CardDAV addressbook server. Hubzilla also has optional stuff like non-federating long-form articles, “cards” that work similarly, a simple built-in wiki engine for multiple wikis per channel with multiple pages each, support for simple webpages (the official Hubzilla website is on a Hubzilla channel) and so forth. I’m not even remotely kidding with any of this.

If you want to unify Fediverse servers, they’d all have to become Hubzilla, but with nomadic ActivityPub.