Why do some countries have "The" often considered as part of their names

I assume various reasons. The Gambia, like The Bronx, was originally named after a river. The Gambia, The Bronx.

OTOH, https://www.youtube.com/watch?v=QcXEGCmCNcI

In three of the four examples you listed it indicates a plural, that is more than one.

The USA and the UK are phrases. Which states? The united states. Which kingdom? The united kingdom, not the one where every fiefdom is fighting each other.

I think The Netherlands are "the nether lands", all the lands below sea level, as opposed to the highlands.

I think the straightforward and boring answer is just because that's what they're called.

There's a vague sort of guideline in English that The Somewhere refers to a geographical region and Somewhere refers to a sovereign state or, at least, defined political territories. Which is why Russia would like us to go back to referring to Ukraine as The Ukraine, and Ukrainians would very much rather we didn't. This also applies to several countries of Africa and some states of India that colonially were The Somewhere and now prefer to be called Somewhere. (Punjab springs to mind.)

But there's no hard and fast rules to these things, and that guideline is not a very well applied at all.