If the letter shown in
this article is accurate, then describing it as "wow look at how cool I am!" or "a message relating to other matters" simply isn't a fair description. It's a direct, straightforward attempt at taking credit for the money being sent to Americans as part of the American Rescue Plan, which Biden did sign into law. It's inappropriate for inclusion in government relief, but is otherwise entirely typical politicking.
Regarding Trump, his name being directly on the checks was essentially designed to cut out the middleman of the government in people's minds. The message is less "the government is sending me money, thanks to Trump," and more "Trump is sending me money." When we receive a check, we generally expect to see the name of the person who's paying us money on it. By putting his name on the checks, Trump was trying to imply - not state outright, because that would be a ridiculous lie he'd quickly be called out on, but imply - that the money was coming from him personally. Even though virtually nobody would actually think to themselves, "Yes, Trump himself is sending me money from his personal bank account," people would still automatically associate Trump's signature with him being very generous with his own money, without explicitly spelling it out to themselves.
I definitely agree that Biden's letter is especially pathetic in light of his administration having made a point of stressing that he wouldn't be doing anything as egotistical as putting his signature on the checks, which I hadn't known about before this discussion. Bragging about their humility could only have been appropriately followed up with the most humble government bailout in history, which this certainly was not. Really, the Biden Administration should just stop comparing itself to its predecessor altogether. It's such a low bar.