Probably. If you don't, then you have two independent variables, and any between-group differences you obtain could be due to either the different drugs or the different routes of administration - with no way to tell which it was.

I guess one exception would be if the way the standard drug is usually administered would be contraindicated for the new drug. For example, some drugs will be destroyed in the stomach, precluding oral administration.

If you think route of administration matters for the new drug, but it isn't the case that it cannot be administered the same way as the standard one, then you might try having three conditions (standard vs. new/given as standard vs. new/given your preferred way). Or, for logical completeness, a 2x2 drugxroute design.

I don't pretend to great expertise here, and stand ready to be contradicted by folks who do this sort of study all the time.

In the assay of the action of new drug against he standard, every condition has to be same including the route of administration. adjustment can come when perhaps when the effect has been established and there is need to consider other dimensions like the mechanism of action.

not necessarily. The aim pursued will be a therapeutic effect, We will compare the achieved effect using a second drug in relation to a first drug (reference) , if we obtain a better result with another route of administration which is also more comfortable, as example oral route instead intravenous, the result will be still more valued (it's an improvement). We can compare effcicacy/safety  and pharmacokinetics (concetrations) results althought the used routes were not the same. By the other hand if we want to do a comparison for a bioequivalent drug, then:  Yes, we need use the same administration route, because we want another drug able to sustitute the reference drug, with the same conditions and results.