I’d say that asking a school to “match” can potentially be problematical, but you can politely ask them to re-evaluate their offer in light of other offers at schools they would be happy to “steal” a student from.
My son was accepted EA Harvard, RD Yale. Harvard’s offer was better. During Bull Dog Days (Yale’s accepted student visit program) I went to the Yale Fin Aid office with the Harvard award letter in hand. I didn’t ask them to “match” but I did say that my son preferred Yale and I hoped that he would be able to make a decision based on the best fit without financial pressure.
The Fin Aid officer told me that she was delighted that I’d brought in the Harvard award letter as it made it much easier for her to increase the award. She asked if she could make a copy of the Harvard award and then she went in to speak to the Dean and came back in less than 5 minutes with an offer equal to and maybe slightly better than Harvard’s. I had an email confirmation within 5 minutes, and a written confirmation letter of the new award was waiting for us when we arrived home a few days later.
About a year after this took place, the Yale Daily News ran an article about negotiating financial aid awards at Yale. https://yaledailynews.com/blog/2016/11/11/how-generous-is-generous-enough/
Son graduated Yale this past May. The award level stayed consistent over the four years.