As another user indicated, most white Americans have a very hard time understanding westward expansion in North America as a colonial endeavor. We inherit a myth of unimpeded westward expansion conveniently omitting the story of the original inhabitants of the land. I noticed this disconnect while reading episode discussion threads for HBO's series Westworld. Hang with me here... In the world of the show, the setting changes from the cowboys/Indians Old West setting to British Raj-based world. It became a widespread joke in discussions on the Westworld subreddit that in the world of the show people would have to be super racist to spend their vacation in a place that glorified British colonialism in India. No one took a step back to see how the previous Old West setting glorified a different type of colonialism baked into the mythology of the American West.

Now, to your question, how was the territorial encroachment across the continent a form of settler colonialism not just territory expansion?

First, and I hate that I have to say this but here we go, there were people living west of the frontier of the newly-formed U.S. It wasn't open space. In fact, up until roughly 1800, there were more indigenous people living east of the Mississippi River than descendants of European colonists.

Colonialism, and specifically settler colonialism, feeds on land. The early United States Indian policy, inherited from a pattern established by earlier settlers, used the threat of genocidal violence to remove indigenous peoples from their homes. A strategy of total war involving the indiscriminate murder of combatants and civilians alike, resource deprivation through intentional destruction of crops, homes, and food stores, and wide-scale indigenous slavery was used to punish those who denied U.S. claims to new land. When outright violence became difficult to justify legal battles went as high as the U.S. Supreme Court, but still forced removals continued, including the Five Civilized Tribes along the Trail of Tears. Many northern nations accompanied their southern neighbors to a foreign land across the Mississippi River. Oklahoma, too, was someone's home. The forced relocation of indigenous peoples in turn threatened to displace nations like the Osage, Quapaw, and Caddo. As more and more settlers streamed ever westward to satiate the need for more land, the United States was unwilling and unable to honor it's treaty promises to prevent settler encroachment across the Mississippi.

Because this land-hungry new nation needed to justify widespread theft, and the violence that accompanied it, the myth of a Manifest Destiny obscured the horrors of settler colonialism. Per the myth, U.S. citizens (read white U.S. citizens) were special, uniquely capable of improving the land, making it useful, and transforming the wilderness into a modern nation. Their claims to territory overrode indigenous claims to the land because they could make it better. Thus justified, white citizens, U.S. troops, and loosely organized local militias could, for example, raze of Navajo/Dine homeland, or hunt buffalo to near extinction, or enslave huge numbers of California's indigenous peoples. Resistance, or even misunderstandings, ended in massacres that would, in popular U.S. history, be remembered as "battles" (Sand Creek, Bear River, Wounded Knee, etc.). In the most sanitized version of this history, we forget there were even people living in the West, and use catastrophic mortality due to infectious diseases as a convenient, and bloodless, excuse to remove indigenous peoples from the narrative.

Settler colonialism requires the organized, very real threat of violence to remove the original inhabitants of a land, and replace them with a new group. The popular perspective of U.S. history ignores this violence laid over the land, and forgets the insidious, ever-evolving methods used to displace indigenous peoples as white settlers moved west. Massacres, legal battles, ever-constricting reservations, forced allotments, forced assimilation, resource deprivation, and enslavement all helped reduce the territory of indigenous people. The story of U.S. expansion isn't white settlers moving into a pristine uninhabited land, but a bloody history we have not fully reckoned with.

Sources for further reading:

Land Too Good for Indians: Northern Indian Removal by Bowes

An American Genocide: The United States and the California Indian Catastrophe, 1846-1873 by Madley

Firsting and Lasting: Writing Indians out of Existence in New England by O'Brien

Surviving Genocide: Native Nations and the United States from the American Revolution to Bleeding Kansas by Ostler

The Other Slavery: The Uncovered Story of Indian Enslavement in America by Reséndez

More can be said, but this answer Why are there so few indigenous peoples in Europe? written by u/Wild_Enkidu touches on the nuts and bolts of what makes colonization "colonization". However it does not provide any comparisons between colonization and territorial expansion

As an indigenous person of North America, I can tell you that American expansion is seen by us in every way as being colonization, and terms like decolonization are ones I hear daily in education and governance. The only specific guide seems to be that if you can get away with being blind to what you did then you might never have to confront the fact that you are a colonizing power whose way of life is built on the exploitation of the land and possessions of others.

To go more indepth, I can see that there might be situations that expansion is not colonization, for example I wouldn't call the Roma spread in Europe colonization, as they were not establishing systems of control over local populations. But in the vast majority of territorial expansions, the concept of control is built right in to the idea - territorial expansion means expanding territory you control, which means either displacing previous controllers, or establishing new systems of power over people who previously had local governance.

Perhaps an example of expansion that might not be explicitly colonization is the example of the Dene expansion southwards in various migrations that resulted in populations in central British Columbia, coastal Washington, and current locations of the Navajo and Apache nations. While some of these expansions definitely involved wresting control of territory from others, archaeological evidence suggests that as a culture they were much better at surviving in certain harsh environments, and mostly moved into vacuums with little or no population, with local populations often intermarrying or being subsumed in. So at least in some parts, this expansion was not a case of colonization (though new research might again show that it was, but the distinction stands). BUT if your expansion involves military, the displacement of people and governance systems, it absolutely IS colonization.