Learn Finnish: Mulla on kaks kissaa: toinen on musta ja toinen on valkoinen. - I have two cats: one is black and the other is white.

Yes, toinen usually means "second" or "other". But when you use it twice in a row like this (toinen ... toinen), it forms a fixed pair meaning "one ... the other". This is the standard way to contrast two things in Finnish.

Could I say yksi on musta ja toinen on valkoinen instead?

You might hear yksi ... toinen in casual speech, but toinen ... toinen is the much more natural and traditional way to express "one ... the other" when you have exactly two items.

Why is kissa in the partitive case (kissaa) but the colors musta and valkoinen are in the nominative case?

The word kissa is partitive because it directly follows a number greater than one (kaks). In the second part, each toinen acts as a new singular subject ("one of them"). When a singular subject is paired with the verb on, its describing adjective (musta, valkoinen) stays in the basic nominative case.

Are mulla and kaks examples of spoken Finnish?

Yes, they are the very common spoken language (puhekieli) forms of minulla and kaksi. It is completely normal in casual written Finnish, like text messages, to see these spoken forms mixed into an otherwise standard sentence.