Your favorite baseball bat deserves to be seen by everyone because it is unique. In fact, your baseball bat display case is sure to look a lot different from anyone thinking to collect them in the 1800s. They didn't have any standardization until 1859 when it was determined the diameter could be no more than 2.5 inches. The shape and length had no restrictions until 10 years later when the maximum length was designated as it is today at 42 inches. In 1884 the Louisville Slugger was first manufactured for a player who kept breaking his bat. The following day, with the new bat, he went 3 for 3.
Not until 1890 did a committee determine bats had to be round and the diameter was increased to the present 2.75 inches. The only difference in major league baseball bats of today and yesteryear are decreased weight and reduced neck size. Ash has always been popular among the MLB players but many have gone to maple after the Barry Bonds feat of 73 home runs in a season with his.
That doesn't mean you can't place that aluminum bat from the minor league or Little League game on a wall mounted display or next to the baseball display stands. Those are quite popular in most leagues other than the majors. The newer bats are stronger and lighter than wood, as well as more durable.
Given the nature of baseballs, you probably don't want to expose the leather to just any old surface. We have baseball display stands and baseball display cases that will protect the shape and integrity of the ball. Stands are great ways to encourage visitors to handle your prize collector's item. If it is old or autographed, prudence might dictate removing it from exploring fingers by placing it in a case, just like your baseball bat display: enclosed but visible for all to see without touching.