(I'm a 19C Americanist by training but material culture stuff isn't my specialty. Apologies for thinnish sourcing.)

One really obvious thing for starters: "the vast majority of Americans" before 1863 includes a lot of slaves, ie people who didn't have legal right to possessions and wouldn't generally be carrying money, wallets, etc.

Keys you'd certainly see plenty of in 1916, but you'd see many fewer in 1816, before the invention of the cheap industrially produced Yale lock. In general again "the vast majority of Americans" wouldn't include very many property owners wealthy enough to secure their property with locks before the advent of the Yale lock in the mid-nineteenth century. Here's a decent-looking source on locks. Before the cheap lock security is basically a part of domestic labor, done either by women or slaves without pay or servants and housekeepers for compensation.

Identification papers in general as an everyday pocket item is largely a mid-20C thing, not earlier, particularly when you're not travelling (it doesn't seem like you're asking about travellers, specifically). Cash, obviously, if you've got it, you carry, but the usual late-20C/21C wallet is full of lots of non-cash identity-tracking items that don't really have any pre-20C analogue. The mid-late 19C got very into calling cards, but even at their height you wouldn't see those in the wallets of "the vast majority," only the middle class.

There's only one real answer to your question I can think of: food and drink. That's the main category you'd see a lot more of in earlier centuries' "everyday carry" than you might expect based on our 21C experience. Depending on what people planned to do during a day they'd be expecting to travel distances with sparse availability of food for purchase, or clean water for the taking. Once you start noticing this you'll see almost any 18C/19C narrative describes bundling up a bunch of sandwiches, etc., as part of one's daily preparations before heading out.

Commenters: Please don't respond to OP's question if all you're going to do is offer speculation. If you have the proper background to be able to answer the question with credible sources, then great. If not, please don't comment. If you do we'll just remove your comments and you're creating more work for us and more frustration for people coming here to see a real answer.

(I really didn't want to leave a sticky comment but people aren't getting the message. I'll unsticky this later.) edit: unstickied

https://www.reddit.com/r/AskHistorians/comments/4avf57/before_we_had_cellular_phones_lets_say_from/

Fairly similar question which may have the answer you are looking for, though the answers are from a slightly later time than you are asking about.

Can I ask a follow up question? When did handkerchiefs go out of style, and is there any discernable reason as to why? Apparently my grandfather always carried one in the fifties, whereas I've never even owned one.