In addition to stroking his own ego, being a "wartime president" is perceived as being an advantage for the incumbent, as voters are seen as less likely to change leadership in the middle of it.
Except this isn't that kind of war. Even if it were, what counts more is how many people it impacts directly. The wars in Iraq and Afghanistan were/are on foreign soil, don't require major impacts on US citizens (e.g., draft, rationing), and still only involve a small percentage of the population.
On the other hand, in Vietnam, thanks to the draft a wider spectrum of the populace was either involved or had the prospect of involvement looming over them. Add to that the untenable situation (the only way to "win" was to escalate and risk nuclear war), and there was enough negative opinion about how the war was being run from both left and right that Johnson didn't run for a second term.
The classic example is FDR, who defied tradition and ran successfully for third and fourth terms. Needless to say, when it comes to quality of leadership, organization, and effective action in the face of dire problems, Trump isn't fit to shine the shoes of the guy who shined FDR's shoes.
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