The coefficients and error terms would almost certainly differ between the state and city level, but for reasons that you probably did not intend.
Coefficients. The way you have posed the problem, you would be taking averages of percentage voting
1 by city without taking population into account. An extreme example shows the danger. Say that there is no racial/ethnic* difference in voting, and that a State has 1000 cities. In the single large city, all 1,000,000 people vote for
1; all 1000 voters in each of the 999 other cities vote for
0. Averaged over people in the State,
1 wins with a bit over 50% of the vote, but
0 gets 99.9% averaged over the cities (without taking population into account).
Error terms. In a linear regression, error terms include everything that the linear model missed: non-linear effects, interactions, important variables missing from the model. So say that the influence of race/ethnicity on voting is different in a rural southern state than a northern urban state. If the model ignores that possibility, then you would expect different errors for within-state versus among-state comparisons. Much of statistics is trying to figure out what lurks within those error terms and to take them into account.
Two messages here. First, although I expect that you intended to take population size into account, the way you posed the model in your question did not. Whether you model your outcome variable as a percentage or as a number of votes, your analysis must include population size. A generalized linear model rather than a standard linear regression is called for.
Second, you need to be a good deal more specific in formulating your model. There are ways to take additional variables, interactions, the fact that each city has its own associated state, and so forth, into account. Even if you choose to develop a simpler model, forcing yourself to consider these additional possibilities will mean that you are at least doing so with your eyes open to them. A solid understanding of different statistical designs, as you might get from an advanced statistics course, would help. See this page as an example of what can be included in a voting/demographic model.
*In US bureaucratic terms, race and ethnicity are distinct concepts. If you conflate them as you do in this question, others reviewing your work will take you much less seriously.