AFAIK, you should get 4 or 5 per character ... When you go to the skill trainer it should tell you how many "tokens' you have ... All players get 1 free one, so if you've already used that one it may say 4.
As a lifetime member I can say that yes, respecs are per character. Each character already level 50 got about 5 repsecs each and characters I created after buying a lifetime sub wound up with a total of about 10 free respecs, some as level up rewards and the rest as veteran rewards.
So when I used a respec on 1 character, the amount available didn't go down for any other characters unless I used respecs on them.
You should get the respecs but as a fellow lifer I found out that you only earn those respecs on characters you are leveling up after the fact. For example if you made a new character and leveled it up that character would get the respecs but the existing level 50 character gets no respecs so you are out of luck.
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So when I used a respec on 1 character, the amount available didn't go down for any other characters unless I used respecs on them.