Not the person you're answering to but the way I interpret their statement is

"my ancestors came here as immigrants in search of a better life 300 years ago. I honor that by helping those doing the same today."

Being American, especially back in 1776 (250 years ago!), was not about ethnicity, but about an ideal - a group of men on the North American continent decided that " that all men are created equal, that they are endowed by their Creator with certain unalienable Rights, that among these are Life, Liberty and the pursuit of Happiness." [0], and then literally put their lives on the line to protect that ideal.

America is, and has always been, a nation of immigrants. What unites it are the common values - in theory, the idea that everyone there has the right to live free and pursue their happiness, as opposed to the idea that everyone should serve the king because that is the will of God, which was the dominant idea in Europe in the 1700s.

With that view, I can see that supporting people who come to the US so they can be free as continuing what the original poster's ancestors started 300 years ago.

[0] https://www.archives.gov/founding-docs/declaration-transcrip...