The argument related to this is multi-faceted. I will explain each component, and then explain the counterargument.

Jordan is Palestinian-Populated

Jordan, as a result of Palestinian refugees from 1948 staying in the country, has had a large Palestinian population. In fact, some estimates have placed the Palestinian proportion of the population at over 50% in the past (and possibly present). This would make the state "ethnically" Palestinian (assuming Palestinian is an ethnic group, which is actually a quite complicated question that scholars argue over), and as such the claim is that Jordan is a Palestinian state by definition.

This claim is weakened by the way that Jordan is run. It is run by a king, who has never been a Palestinian himself (in the history of Jordanian kings). If the state were a democracy, Palestinians would have had a distinct chance at running the state, and thus would have been a "Palestinian" state where Palestinian self-determination could be exercised. However, this has never historically been the case.

Jordan was part of Palestine

This argument rests on the pre-1920s definition of "Palestine". As a geographic region, Palestine was never administratively defined by the Ottoman Empire with distinct borders. However, it is very likely that popular consideration of what the region "Palestine" was also included at least part of Jordan. This is made somewhat clear by the British Mandate granted by the League of Nations. In Article 25, it says:

In the territories lying between the Jordan and the eastern boundary of Palestine as ultimately determined

"The Jordan" refers to the Jordan river, which means that Palestine had uncertain boundaries and thus extended into Jordan. Indeed, for around 8 months, Palestine included (in the British Mandate for Palestine) Jordan, until it was separated rather arbitrarily. The initial thinking, many argue, was that the Jewish state would be west of the Jordan river, while the Arab-Palestinian state would be east of the Jordan river. The thinking goes that this was not implemented with the change of heart engendered by the Churchill White Paper, which said that the Balfour Declaration did not entitle Jews to "Palestine", but rather to a national home within Palestine, which meant divisions of it.

Of course, whether or not Jordan was part of historic Palestine means little in the grand scheme of things to Palestinians today. Again, they seek self-determination on the basis of Palestinian nationalism, and Jordanians (who might once have been considered "Palestinian" under the British Mandate in the early 1920s, or earlier) do not consider themselves Palestinian. Palestinians have a national movement, whose goals are debated, but there's no debating that Palestinians are not subscribing to the view that their national aspirations are satisfied by Jordan's existence, and do not regard Jordan as a state of their own.

Jordan Should Be Palestinian

This argument comes primarily from some groups, among them Israeli politicians and formerly powerful Palestinian militant organizations. The king of Jordan would also show solidarity with Palestinians by claiming "Jordan is Palestine and Palestine is Jordan", which fed into this narrative. Jordan, of course, sought to control the West Bank, but formally gave up its claim before the turn of the 21st century for various reasons, including that it remained unlikely it would ever reoccupy the territory. Pan-Arabists like George Habash of the Popular Front for the Liberation of Palestine (PFLP) would often claim that "the road to Tel Aviv runs through Damascus, Baghdad, Amman, and Cairo", though the Amman phrase was quite common. The PFLP and other groups believed that Jordan, and other Arab monarchies, stood in the way of Palestinian victory over Israel, but the slogan is sometimes distorted to mean that Amman should become Palestinian-controlled to allow Palestinians to destroy Israel and "rejoin" the historic region of "Palestine".

That's a brief overview. If you want to hear the arguments that Jordan is Palestinian as written by some, you can read the claims by Raphael Israeli, in his article "Is Jordan Palestine?". You can evaluate the arguments with the basis I've given you, and Google can help explain more counterarguments and arguments about the topic. But the claim that Jordan is a Palestinian state remains fringe.

This might be something I can contribute to. To answer you question, it depends on your definition of Palestinian, (and maybe your definition of what a "state" really is") but probably no, Jordan can not be considered a Palestinian State.

To make matters more complicated, there really isn't a precise definition of what "Palestinian" really is, but most people can agree on some features Palestinians share: Arab descent, living within the borders of what later become Israel (pre-1967) and later, where the sentiment you have echoed with Jordan probably comes from, in the West Bank and Gaza. The West Bank, pre-1967, was in fact, part of Jordan, Transjordan as it was still called back then. You really have to go deep into Jordanian history to understand the details about why the West Bank is not considered a part of "proper" Jordan, but I'm giving a quick overview. The ruling family of Transjordan at that time were Hashemites (even to this day, it is still called "The Hashemite Kingdom of Jordan"), but they were not Jordanians. Transjordan was basically given to King Abdullah I as a reconciliation prize by Winston Churchill and Western Powers, after he the newly-found State of Iraq threw him out (There is some contention about this, Abdullah always claimed he didn't really want to be King of Iraq). So Transjordan had this royal family transplanted onto them, with barely any connection to the land and no backing within the population.

Jordan held onto the West Bank of the Jordan River until 1967, when they lost the war against Israel, only holding on to the East Bank (and of course, other parts farther east). A bunch of people living in the West Bank were former Jordanian Citizens, and in fact, Israel treated them that way for a long time (basically until the first Intifada). While there were some native Jordanians living in the West Bank between 1948 and 1967, a large part of the population where refugees from the 1948 Independence War and also didn't gain statehood with the Transjordan Kingdom. The Hashemite Family used the justification that integrating them into the Transjordanian State would be to admit defeat and recognize Israel and that they lost the war. The long game for all Arab states during this time was still the destruction of the state of Israel, it wasn't until Sadat (of Egypt) signed the Peace Accords that they recognized the status quo somewhat. This doesn't really have relevance to your question, but I'm just giving some background.

So the people in the West Bank are stateless actors, with Jordan not giving them citizenship but Israel treating them like Jordanian refugees. This is really when Palestinian nationhood started to emerge, with Arafat and the PLO. They were based out of Jordan until the events of Black September, 1971. This was when the Hashemite Family threw out the PLO in a sort of civil war. The reasons and justifications about why they did that are heavily debated until this day, but the effect was the same for the Jordan Kingdom: Palestinians felt betrayed and did not want to be represented by Jordan (even less by a Hashemite ruling family which wasn't even Jordanian). This is really where the two paths separate (quite violently) and we can begin speaking about a distinct "Jordanian" nationality and a "Palestinian" nationality.

It is true, though, that some suggest that Jordan should just integrate the West Bank again (under some sort of Peace Agreement with the PLO/Hamas) and then there would be peace in the region. Sometimes this has been advocated especially by Israeli politicians (this was more often pre-1980's, when it was still an Arab-Israeli Conflict, not an Israeli-Palestinian Conflict). Integrating Palestinians would probably be heavily against the wishes of the Palestinians themselves and also of the Jordanians (who see Palestinian refugees as poor, uneducated and a burden on their fragile system). This point is not to be underestimated, as the recent conflicts (last 20 years) have shown, Palestinian feelings about not being represented properly are some of their biggest grievances.

I hope I have somewhat answered your questions. If there are any mistakes with the language, please correct me, English is not my first language.

Sources: Tessler, Mark (2009): A History of the Israeli-Palestinian Conflict, 2nd Edition [pages 399-534 deal with the Jordanian question)

Smooha, Sammy (1989): Arabs and Jews in Israel: Conflicting and Shared Attitudes in a Divided Society. [Smooha is one of the better researchers in the area, and he has gone to considerable lengths to ask Arab Israelis and Palestinians living in the West Bank how they feel about nationhood, statehood, and what they identify with. He has done these surveys for over 20 years now and is one of the most respected scholars when it comes to the Palestinian statehood question.]

The British mandate was made up of two areas, Palestine, which is slightly smaller than modern Israel, and Transjordan, which became parts of Jordan and Syria. During the Byzantine period the region was made up of provinces Known as Palaestina Prima, and Palaestina Secunda. Before that it was the Roman Provence of Syria Palaestina. There was also never a country called Palestine, what are now Palestinian claimed areas were administered by Jordan (east Jerusalem and the west bank).

After the British mandate and some of the early wars of the state of Israel (basically the 1947-1967 period) many Palestinians resettled in Jordan (and other Arab countries) but Jordan was the only country to fully naturalize Palestinian refugees.

So at times parts of what is now Jordan and Syria were called Palestine, and many Palestinians live in Jordan. Jordan also administered what is now called Palestine for a while. Saying more than that with a situation as complicated as exists over there is difficult, and I'll leave it for somebody better educated than myself.

Edit: doing some more reading (yay wikipedia sources) Found the Minorities at risk project assessment of Palestinians in Jordan it said:

Palestinians are not distinct from Jordanians, but differ in historical origin

full report here: http://www.cidcm.umd.edu/mar/assessment.asp?groupId=66302

Edit Two: Full disclosure I may be biased on this subject; although I was born and raised in the US I hold Israeli citizenship and lived there for several years.