τρώτε or τρώγετε ---> Second plural person Imperative in Present Simple (Enestotas), Active Voice
It can be also Second plural Indicative person in Present Simple/Present Continuous (Enestotas) = You eat, You are eating
φάτε --- Second plural person Imperative in Past Simple (Aoristos), Active Voice
As about the brand name ΦΑΓΕ it probably comes from the 2nd singular Imperative person in Past simple (Aoristos), Active Voice of the word τρώω = eat, which is φάε or φάγε. Some say it's just an acronym without any other meaning.
Just like in the subjunctive (see here), the verb tense in the imperative actually carries no time information, only duration information.
- Τρώτε/τρώγετε: present tense of the imperative, therefore continuous/habitual action (plural); singular: τρώγε
- Φάτε: simple past tense of the imperative, therefore instantaneous/happening once action (plural); singular: φάε
- At a children's party, the hosting mom says: "Eat your food and then we'll play" = Φάτε το φαγητό σας και μετά θα παίξουμε. "Eating" here is perceived as one of a series of actions happening once, hence simple past of the imperative.
- Two little children play and talk instead of eating. This has been going on for quite some time, so the father says: "Eat your food quickly!" = Τρώγετε το φαγητό σας γρήγορα! If there was only one child delaying: Τρώγε το φαγητό σου γρήγορα! "Eating" here is a continuous action, hence present of the imperative.
Ok so let's say I have a mouthful of chocolate (I can eat it all at once) and somebody wants me to eat it, then he would have to say "Φάτε την σοκολατα" because it is just going to take a few seconds. But if a have a plate of pasta (nobody knows how long it could take me to eat the whole thing), and somebody wants me to eat it, he would have to say "Τρωετε τα ζυμαρικά"
Similarly, if there are a lot of mouthfuls of chocolates that somebody wants me to eat he would have to say "Τρωετε την σοκολατα" because the action will be repeated several times
Am I right?
No, in both cases described there is an instance of the action. You are going to eat some pasta or chocolate today and that's it. Τρώτε (not τρώετε) would be used in a sentence like παιδιά, τρώτε το φαγητό σας=kids, eat your food (in this case, this is a general command you will have to follow repeatedly. We are saying to the kids to eat their food every time they have a meal). Παιδιά, φάτε το φαγητό σας=kids eat your food (in this case, we are saying to the kids to eat this instance of food that we have - could be a plate of pasta, could be two or more plates; that depends on context)