Part of the answer is that Lincoln had to be a shrewd politicians and play to his audiences when running for office. His private and publicly expressed beliefs did not always match.

The next part of the answer is one of presentism, a rather pesky problems for historians whereby an observer in the present day inadvertently projects their own values onto the past instead of considering the past within the context in which it took place. Simply put, our notions of equality did not exist back then except among a very select group of people (usually abolitionists or the most radical of Republicans, neither of which garnered widespread Northern support in the antebellum era). Most often, equality for Northerners like Lincoln often was based on control of one's labor and the ability to achieve social mobility through hard work.

What Lincoln often expressed was that even black Americans deserved the right to possess and profit from their own labor, as opposed to having a slaveowner profit from it as often took place in the South. The ability to determine one's social position and prosperity through labor and the fair acquisition of money and property was seen by many Northerners as the essence of American liberty, and even blacks deserved to have their shot at it. Of course, that didn't stop many Northerners from still harboring highly racist and unequal attitudes toward blacks, slave or free, in terms of political, legal, and social rights. Often Northerners advocated the rights of blacks to control their own labor as a means of trying to shore up their own personal rights to the same thing. But if black Americans were to have those rights, Northerners still wanted to keep them in the South and out of the North and West, or preferably out of the US altogether. This is why many Northerners, including Lincoln, advocated what was called colonization efforts, by which free blacks (and potentially freed slaves) were encouraged or (hypothetically) forced to leave the US for some Caribbean or African location.