The exact limits of Presidential impeachment proceedings have not been tested in courts. They've barely been tested at all. Right now is only the fourth time impeachment inquiries have reached any level of significance (meaning, not just rhetorical bombast) in the country's history. The first time there was no court involvement at all that I'm aware of (beyond Chief Justice Salmon Chase's mandatory appearance in the Senate Trial, and the Senate frequently over-ruled whatever he tried to do), and even the President's closest advisers willingly and openly talked to the House in great detail. The second time occurred much later with Nixon, and most of the court precedents we got from that weren't directly concerned with the impeachment proceedings but rather with criminal investigations, including the ruling on the tapes. The third was Clinton, and much like with Nixon most of that was actually conducted by a non-Congressional investigator; the House itself had to do very little other than rely on what was provided to them by Starr. So right now is the only modern incident of an impeachment where the investigations are primarily executed by the House itself, and the nature of the Presidency has changed dramatically since the first time around.
If we expand matters a little more, to impeachment in general, then what little the courts have said (as there are still few cases even then) is that they consider the use of the word "sole" in describing the House's possession of the power of impeachment, and for the Senate's possession of the power of trying an impeachment, as extremely significant. That word is used nowhere else in the constitution. Combined with the fact that impeachment is also a power wielded against the Judiciary to check against any abuses by the Judiciary means that courts have been largely unwilling to touch the power of impeachment. They deem that it would be improper for them to effect constraints on a power that is used against them in the system of checks and balances.
And more generally, the powers of Congress that resemble the powers of the Judiciary, such as subpoenas and contempt and otherwise forcing testimony, are not treated identically to those powers.
A notable case is the notion of attorney/client privilege. This protects, in normal court proceedings, communications between an attorney and a client that cover a real, articulated legal need; the idea being that it allows the client to be fully honest with their attorney, which is necessary for formulating the best possible defense, without it endangering the client. This power does not exist before Congress. When Congress respects it, it is the exercise of a political choice. So Congress could, in fact, subpoena an attorney such as Rudy Giuliani and compel him to speak on matters that would be (alleged to be) covered by attorney/client privilege in a normal court. Though note my description of this privilege should make it clear that it does not cover nearly as much stuff as people like to think it does when talking in the media, including Rudy himself.
The principle defense that is invoked in an attempt to defy subpoenas is based on a combination of executive privilege and separation of powers. The executive privilege angle is something of a recent addition historically speaking, usually as a support to the bigger separation of powers argument. The privilege is basically a variation on the attorney/client privilege, viewing open and protected communications as essential for the functioning of the Executive branch, only applying more broadly (and not appearing anywhere in the Constitution; it's a legal fiction deemed necessary for what is intended or explicit).
It is largely accepted that Congress cannot compel the President himself to appear before them and provide testimony, nor can they punish him for failure to do so (or do so honestly) beyond impeachment. He exists as co-equal to Congress, and so except as expressly detailed in the constitution they cannot compel each other to do things. And the courts have, to date, been increasingly willing to consider the President's closest advisors as extensions of the President, who thereby similarly enjoy the same protections in this regard. President Trump is currently arguing to vastly expand this protective umbrella to virtually anyone the President talks to, or serves within the Executive Branch. We would have to simply wait to see if these arguments go before SCOTUS, and what they say about it.
As such the resolution to this problem is to resolve what happens when the "sole power of impeachment" clashes against "separation of powers and executive privilege", and how far one can intrude upon or arbitrarily defy the other. Personally, I think that will be the most important precedent that might yet come out of this whole affair. But it's not even sure it will get there. The House currently does not seem willing to wait for the courts to sort this out over the course of months or more, as then the political landscape may change drastically and unpredictably. Plus, the aftermath of a loss for either side could be drastic, both immediately and far into the future. So the matter may become moot before it gets to SCOTUS, and as such they may simply not end up addressing it.