It really depends on the school. In general, popular state flagships tend to have the most issues with more students wanting to major in engineering than the departments have space for, since they have plenty of interested students who are strong enough to handle the rigor of an engineering major.
A. Some fill each major with direct frosh admission. But then there is little space for those who want to change majors later, or enter an engineering major after being non-engineering, so doing so can require a very high college GPA or highly competitive process. UT Austin, UIUC, and many of the UCs are this way.
B. Some admit to a pre-engineering program, then have frosh/soph students compete for majors by college GPA and sometimes essays. Often only a few majors are competitive. Examples include Texas A&M, Minnesota, Purdue, Virginia Tech. A variant is the setting of progression GPAs that may be higher than 2.0 to avoid dismissal from the major, like at Wisconsin.
C. Some have plenty of departmental capacity to allow any student in the engineering division to choose any engineering major, with just a 2.0 GPA in prerequisites. Many less selective schools are this way, because the rigor of the curriculum weeds out students so that fewer choose engineering than the departments have room for. Among more selective schools, the best funded ones can be this way. Michigan is probably the best known public example; private ones would be Stanford, MIT, etc…
What you may want to do is web search for “[school name] change major [desired engineering major]” to get an indication of how “full” each engineering major at the school is.