No, Summon Greater Demon is a specific rule that applies to that spell.
The Rule of that spell does not transfer to any other spell without that text.
You may issue your verbal commands for conjured beings at any time. They will execute your commands based on the commands you issue: Say: attack X, and it will attack X. Say defend yourself until X appears, then attack X and they will defend until X appears, a new command is issued, or they disappear.
If you don’t issue any commands to them, they defend themselves from hostile creatures, but otherwise take no actions.
Conjure Fey and Conjure Celestial also have the restriction that your commands are not obeyed if your commands violate their alignment.
as long as they don’t violate its alignment
See Conjure Animal
The summoned creatures are friendly to you and your companions. Roll initiative for the summoned creatures as a group, which has its own turns. They obey any verbal commands that you issue to them (no action required by you). If you don’t issue any commands to them, they defend themselves from hostile creatures, but otherwise take no actions.
You do not have to issue a new command to the conjured being each turn in order for it to follow its command on its next turn.
See XGTE, Summon Greater Demon, Page 166:
When you summon it and on each of your turns thereafter, you can issue a verbal command to it (requiring no action on your part), telling it what it must do on its next turn. If you issue no command, it spends its turn attacking any creature within reach that has attacked it.
At the end of each of the demon’s turns, it makes a Charisma saving throw. The demon has disadvantage on this saving throw if you say its true name. On a failed save, the demon continues to obey you. On a successful save, your control of the demon ends for the rest of the duration, and the demon spends its turns pursuing and attacking the nearest non-demons to the best of its ability.
This is especially important because conjured beings are friendly while demons are (almost every time) hostile. This is an issue of being in command and has severe consequences.
A conjured animal will stay friendly without you issuing commands on your turn. A summoned demon will act on its own.