The FCC by law cannot have more than three commissioners from the same party. The sitting President therefore finds himself in the position of having to appoint up to two commissioners who are NOT from his own party.
I don't think that there is any actual requirement for how the President picks such nominees--so a Democratic President needing to appoint a non-Democrat could in theory choose someone from some left-leaning non-Democrat party like the Green party, and a Republican President could choose someone from a right-leaning non-Republican party, such as the Libertarian party.
In practice, though, what Presidents of both parties usually (always?) have done is ask other party leadership [1] for a name, and then the President nominates that person.
When the nominee comes up for confirmation in the Senate, generally the Senators from each party pretty much automatically vote to confirm the nominees from the other party unless there is something that actually disqualifies them. They don't vote no just because they disagree on ideological grounds.
So yes, Obama originally put Pai on the FCC, but you can't really read anything into that as far as Democrat positions goes. Pai was the choice of Republican leadership for one of the two seats that could not go to a Democrat.
People are too focused on Pai here. Getting rid of net neutrality is in the freaking GOP party platform. By winning the White House, Republicans won a majority on the FCC. It didn't matter which existing Republican commissioner they elevated to the chairmanship (Pai or O'Reilly) or if they made their new, third guy chair (Carr). Whoever they picked was going to do this.
[1] Usually whoever leads the other party in the Senate, I believe.
As for "voting in lock-step", nope not even close. Good analysis of many major votes here: https://www.reddit.com/r/cantmisslists/comments/7gaq5z/both_... Democrats vote to keep the government transparent, honest and benefiting the people way more than Republicans.
From where I sit, the particular observation you're making about how policy has been treated when it comes to Net Neutrality issues looks exactly like how the Republican party behaves generally. Whether it's about net neutrality, weapons of mass destruction in Iraq, climate change, stimulative effects of tax policy, health care -- it sure looks like many Republican positions are primarily arrived at and founded in the profit and power aspirations of a narrow constituency rather than observation and study, and that "deregulation" and "market-based" approaches are primarily invoked as tools or even just fig leaves where they appear useful.
Net Neutrality just happens to be one example where the audience here is predominately familiar enough with the relevant practical issues that it's easy to see.