Your first use of Favored Foe takes your concentration, and this will immediately drop your concentration on hunter's mark, as per the spell concentration rules and the clarification of those rules in the SAC. However, your first use of Favored Foe starts "when you hit", so you have a number oftwo other immediate consequences as well: your base damage, your Favored Foe damage bonus,(applied "on a hit") and your hunter's mark damage bonus (which is applied "whenever you hit"). Thus your one event, hitting, triggers fourthree different consequential effects - and it is up to the DM to decide the resolution order of these simultaneous effects. As soon as either of the two damage effects resolves, you also trigger another immediate effect - the Favored Foe damage (which occurs the first time on a turn you "hit...and deal damage"
Thus, your DM can rule that the resolution order of the four effects after the hit is: establishmark your Favored Foe, do base damage, do Favored Foe bonus damage, do hunter's mark bonus damage, concentration drops on hunter's mark.
Your DM could also rule that the resolution order is: establishmark your Favored Foe, drop concentration on hunter's mark, do base damage, do Favored Foe bonus damage, no hunter's mark bonus damage is done because at the time the hunter's mark bonus damage (triggered by the hit) resolves, there is no longer a hunter's mark in effect.
These events can be resolved in any order the DM decides, with the only limitation being that the Favored Foe bonus damage cannot occur until after one of the other damages has resolved. In particular, Therethere is no priority given to dropping concentration effects occurring before the other consequences of the hit, unless your DM decides to do so.
Your DM could also decide to use the XGtE optional rule, which allows the player whose turn it is to decide the order of resolution of simultaneous events. In this case, you as the player of the ranger can certainly arrange them so that you get the bonus damage from hunter's mark once before your concentration on it drops.
RAW it is unclear
The spellspell concentration rules are pretty clear - you cannot concentrate on two thingsspells at once, so as soon as you begin concentrating on somethingone, you lose concentration on anything elseany other on which you were concentrating. So the question is, does Favored Foe, not being a spell, actually require concentration? That depends on what the phrase 'as if' is modifying.
"As if" could be modifying "your concentration". That is, it could be saying: You are not actually concentrating on a spell - but this class feature requires concentration like a spell does, and thus anything that breaks concentration will result in the early end of this effect. Also, you are effectively concentrating for the purposes of other effects that interact with concentration. If this is the intended interpretation, then using Favored Foe will immediately end your concentration on hunter's mark. (as is assumed in the discussion above about simultaneous effects).
On the other hand, "as if" could be specifically modifying "lose your concentration". That is, it could be saying: You are not concentrating on a spell, and you are not even concentrating on the class feature - not in any sense that the rules define as 'concentration'. However, this class feature can be ended before its full duration by any effect that would cause you to lose concentration on a spell, as if you had been doing so. In other words, you do not actually concentrate to use the feature, but things that would make you lose concentration will make you lose itthe feature, too. If this is the intended interpretation, then using Favored Foe will not end your concentration on hunter's mark. Casting hunter's mark while you had Favored Foe running, however, would cause you to lose Favored Foe - you can lose Favored Foe, but using Favored Foe does not make you lose anything else.
There is no way, RAW, to decide which interpretation is correct. because the language of Favored Foe is not clear.
While all of these features are examples of "as if" concentration without spellcasting, none of them can help decide this question because they use the same or similar language. They all beg the same question of whether you are actually concentrating in a positive sense, or are just saying that the feature can be ended in the same ways that concentration is ended.
Even more informative is a ring of djinni summoningring of djinni summoning, which says the summoned djinn "remains as long as you concentrate (as if concentrating on a spell), to a maximum of 1 hour..."
Here we see a very clear statement - you actually are concentrating, just as if you were concentrating on a spell. This language is also used in the *ring of shooting stars*ring of shooting stars: "The spheres last as long as you concentrate (as if concentrating on a spell), up to 1 minute."
So, RAW, we know how Visions of the Past and the two rings work, and we know that using any of them would end hunter's mark. Can we infer that all the other features, including Favored Foe, do the same? Only if you want your rulings to make sense. Only if you We can if we value coherency. RAW, Favored Foe is vague and requires DM resolution. If you think that all non-spell concentration effects should work the same as each otherthe same as each other, then by RAI Favored Foe acts in all aspects like youryou were concentrating on it. With the text of Favored Foe as written, your DM could rule that using it does not even break your concentration on hunter's mark, but if they like to be consistent in such rulings, it should break concentration because other similar features do.