Someone played EUIII a little too much

. In all seriousness, if Poland, or the Polish-Lithuanian-Russian Commonwealth, managed to hold all this land then there is nothing but a couple of weak khanates between it and the Pacific. Russia never really conquered anything east of the Siberian Khanate until it hit the Qing, kind of like how Canada never really "conquered" its western portions, it just sort-of happened.
That's a bit of a misconception. There was definitely some conquering involved and the government was involved in the conquering.
If anything, the Spanish effort was even more individual-driven than the Russian effort. Russia's cossacks and private explorers were followed up by regular troops and support logistics from the European part of Russia almost immediately, and since it was overland it wasn't cheap. Local notables also enrolled in the Russian army and administration and got salaries in flour, sugar, tea and cloth - where do you think that came from? The cossacks in their forts certainly couldn't produce all that. They were brought from Europe.
That said, it was still a shoestring kind of conquest, made easy by the fact that the Russians came in with full knowledge of how yasak economies operated and simply took over from the Bukharans, Taibughins and Yakuts.
I don't see why it would not be very similar even if Poland nominally conquered Russia. It would probably be the same Russian explorers, and the regular troops sent after them could perhaps have a Polish officer among them now and then.
The Spanish eventually ran into the Chichimeca wars, the Mapuche wars, the long-lasting conquest of the Maya, the Comanche raids - the closest equivalents Russia found were the Chukotka wars and some raiding by the Nogais. The Kazakhs were generally friendly and the Oirats were generally allies, so the 17th c. phase of Russia's expansion as compared to Spain's went much more smoothly, where the 16th c. was more comparable.
Why would Poland even need Siberia?
Same reasons Russia did?
1. 15th/16th c. - the silver tribute. By the mid-16th c. the local people aren't trading for silver with Central Asia anymore so the silver tribute dries up, but then there is the...
2. Soft goods, i.e. furs, collected by yasak. This was hugely lucrative, and brought the Russians all the way to California, just hunting those furry critters down.
3. Salt. Northern Urals have a lot of medieval and late medieval saltworks. So does the trans-Ural area. This gives you a population base from which to diversify Ural and West Siberian economies, which leads to...
3. Mining! Urals have copper, iron, and lots of precious and semiprecious stones. By the 18th c. there were dozens of factories all absorbing people from European Russia to work there....
4. ...which leads to a pretty self-sustaining economy with a set of imports, exports and by-products. By-products are things like hides, leather, livestock, metalwork. Combine it with a little of the fur trade and you have something that...
5. ...China wants. So 17th c. onwards saw a fair amount of trade overland through Kiakhta, various Russian goods in return for mostly tea and a few other items. It was also quite profitable.
So there you go. There are lots of things to do in Siberia and it doesn't cost that much in terms of national manpower to develop these things.