Would the German Emperor talk to the President of France respectfully like to the Queen of Great Britain or more like to the Prime Ministre? And vice versa, would the republican leader have to accept monarchistic conceitedness due to republics being so outnumbered by monarchies back then? And how did contracts between monarchies "by the grace of God" and republics work?
Not to foreclose any other answers, but as a starting point, you might consider my answer from last year (one day shy of exactly one year ago, in fact) about the privileges that elected heads of state received when visiting European monarchies in the late 19th and early 20th centuries.