This article explains the procedure behind it. A few quotes:
Electoral fusion is an arrangement where two or more political parties on a ballot list the same candidate, pooling the votes for that candidate. Distinct from the process of electoral alliances in that the political parties remain separately listed on the ballot, the practice of electoral fusion in jurisdictions where it exists allows minor parties to influence election results and policy by offering to endorse or nominate a major party's candidate.
Electoral fusion was once widespread in the United States[...]. By 1907 the practice had been banned in 18 states; today, fusion as conventionally practiced remains legal in only eight states, namely:
- California (Presidential elections only)
- New York
- South Carolina
From my POV, I can see two ways of valoring the situation:
As a way of independent political organizations to get a say in national politics (they may not be strong enough to present a candidate, but may force one of the candidates to negotiate some proposals in exchange of their support).
As a way of already stablished candidates (Reps and Dems) to get a "second brand" that caters to people too extreme for the mainstream parties.
Imagine1 that there is a large enough group of people that say that to get voting rights you need to know how to play a musical instrument; a mainstream candidate appealing for their support could lose a lot of votes elsewhere. So, instead of the mainstream candidate X making campaign for that demographic, a new party appears that includes as part of its platform promises to get that objective, and that such a party then endorses candidate X... the "second brand" gets the votes without X having to compromise his/her position.
From what I read, some parties look more like 1st (not always endorsing the same candidate, and some times endorsing none) and others may be more suspect of using the 2nd possibility2.
And, as for the last question "is Trump elected as a Republican President or this particular party's President?", the answer would be:
Presidents may belong to a party, but what is elected is the candidate, not the party. So, formally it would be "Mr. Trump the President" and not "Mr. Trump the Republican President". Anyway, as telling which party the candidate runs for gives a first idea about his/her political views, that information usually comes together.
You could say that Trump would be the "Republican & American Independent Parties President", but the other parties are so small and little known in relation to the mainstream parties that it would give no practical information.
1To avoid second guessing I am going to use a completely hypotetical and extreme situation, in case you don't notice it.
2Although it is hard to be sure just by looking at their support history, some party supporting always the Democrat or Republican nominee does not imply that they are "second brands" of these parties; it might simply mean that the party position is too much to the left or to right to have another viable endorsement opion.