Wording is sometimes hard and upon first glance can appear to be contradictory. For instance, Brimaz, King of Oreskos allows for its token to attack as a third combatant when cards like crawlspace are in play.
Cards like Crawlspace and Silent Arbiter create attacking and/or blocking restrictions. Those only apply to, and are only checked during, the turn based actions of declaring attackers or blockers. They have no effect on the game at any other time. In particular, they don't affect creatures that enter the battlefield attacking/blocking or are made into attacking/blocking creatures after the turn based action has already happened.
So for your question, there is no interaction, Tahngarth's second ability doesn't care about the Arbiter's effect.
508.4c A creature that’s put onto the battlefield attacking or that is stated to be attacking isn’t affected by requirements or restrictions that apply to the declaration of attackers.
509.7b A creature that’s put onto the battlefield blocking isn’t affected by requirements or restrictions that apply to the declaration of blockers.
Sadly, the rule team were a little short when they wrote rule 509.7b; it should mimic 508.4c. Let's hope they'll correct this oversight in some future rules revison.
508.4c A creature that’s put onto the battlefield attacking or that is stated to be attacking isn’t affected by requirements or restrictions that apply to the declaration of attackers.
509.7b A creature that’s put onto the battlefield blocking isn’t affected by requirements or restrictions that apply to the declaration of blockers.
Sadly, the rule team were a little short when they wrote rule 509.7b; it should mimic 508.4c. Let's hope they'll correct this oversight in some future rules revison.
Currently the only effect in Magic I could find that provides that a creature "is blocking" another when it formerly wasn't is found in General Jarkeld (C.R. 108.1), and the reason for this text is to avoid the word "switch" in Jarkeld's former Oracle text, not necessarily to introduce a novel game mechanic. On the other hand, the rule change in C.R. 508.4 and 508.4a in Commander 2019 occurred primarily because of Tahngarth, which introduced a novel game mechanic in that set with an effect that states that a creature "is attacking".
EDIT: Edited slightly, after comment 6 was posted.
Blocking is different from attacking. Blocking subsumes the complementary "blocks/blocked-by" relations, whereas attacking is truly just an action that occurs as part of the turn structure or a directed transformation as on Tahngarth, changing a single permanent (creature). It may remain the intent of WotC to have the requirements and/or restrictions that applying to blocking remain in force even for future General Jarkeld imitators.
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So, my question is: how does the wording on Tahngarth, First Mate interact with cards like Silent Arbiter ?
So for your question, there is no interaction, Tahngarth's second ability doesn't care about the Arbiter's effect.
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509.7b A creature that’s put onto the battlefield blocking isn’t affected by requirements or restrictions that apply to the declaration of blockers.
Sadly, the rule team were a little short when they wrote rule 509.7b; it should mimic 508.4c. Let's hope they'll correct this oversight in some future rules revison.
RULES OF MAGIC :
http://magic.wizards.com/en/game-info/gameplay/rules-and-formats/rules
EDIT: Edited slightly, after comment 6 was posted.
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