Eldritch blast uses the Cast a Spell action. Making attacks uses the Attack action. A character with extra attacks from the Extra Attack feature doesn't gain the ability to cast more spells.
However, a level 2 fighter can use the action surge feature to obtain a second action in a turn one time/short rest. This action could be used for the Cast a Spell action or the Attack action among others. So one time/short rest a warlock X/fighter 2+ could use action surge to cast Eldritch blast twice or take the attack action twice (or one of each for that matter) on a turn.
Could a warlock/fighter fire multiple eldritch blasts in one turn because you get extra attacks as a part of fighter?
No.
The fighter gets its extra attack from taking the attack action. Eldritch Blast is the cast a spell action.
You can get extra melee attack from being a fighter 5. You can get extra eldritch blasts from hitting class level 5, but they don’t stack.
Now, your fighter action surge would let you use eldritch blast twice in a round, but only one time per rest.
Eldritch blast uses the Cast a Spell action. Making attacks uses the Attack action. A character with extra attacks from the Extra Attack feature doesn't gain the ability to cast more spells.
However, a level 2 fighter can use the action surge feature to obtain a second action in a turn one time/short rest. This action could be used for the Cast a Spell action or the Attack action among others. So one time/short rest a warlock X/fighter 2+ could use action surge to cast Eldritch blast twice or take the attack action twice (or one of each for that matter) on a turn.
As stated above, extra attack applies to the attack action, which casting a spell is not (even if the spell includes an attack).
You can cast it a second time using action surge though, but that has a limited number of uses.