It's true that there are just as many odd numbers as even (using most reasonable ways of counting; things always get a bit dicey with infinite sets), and just as many multiples of 7 as "3 more than a multiple of 7" and so on.
Still, there's a good reason to privilege the multiples. With regular addition of the integers, the number zero has a special role, in that n + 0 = 0 + n = n for all n. It's called the "additive identity", and it's the only number that has this property. If we think of inverses of numbers, like "what's the opposite of 19?", then in the world of addition, they are defined in relation to 0. The "opposite" of 19 is -19, because 19 + (-19) = 0.
Many algebraic structures have an identity; in the world of multiplication of fractions, the identity is 1, and the inverse of 19 is now 1/19. A more abstract example would be the operations on a Rubik's Cube, where the identity is "do nothing". That's the least exciting thing to do with a Rubik's Cube, but it has a special role, just like 0 with addition. If we want to talk about inverses of Rubik's operations, then again, they are defined in relation to the identity: the opposite of "rotate the top face a quarter turn clockwise" is "rotate the top face a quarter turn counterclockwise", because the sequence of those two operations gives you "do nothing".
It is in this sense that "multiples of n" are special, because they effectively comprise the identity element under addition modulo n. That is, if we add numbers and only look at the last digit (in other words, the remainder after dividing by 10), we'll find that adding 0, or 10, 20, 30, etc., leaves that digit unchanged. Another way to say this is that if you take two numbers with the same last digit, their difference will be a multiple of 10.
In other words, it isn't merely that there are just as many numbers in one set as another, it's that one of the sets acts as a point of reference. For a real-world metaphor, consider the concept of birthdays (disregarding complications like leap years). If you were born on February 5, then every other February 5 is a birthday, because the difference of those two dates is a multiple of 365. This might highlight the conceptual argument: I would agree that there's nothing fundamentally more special or interesting about February 5 than August 27 or any other day, but it's when we start comparing dates or using them in some frame of reference (like trips around the sun) that the number 365 and its multiples come into focus.
Or, for a real-world example related to evenness vs. oddness, go and flick a light switch an even number of times. If the light was off to begin with, it will still be off at the end; if it was on, it will still be on. Now, if you have a fancy lamp with three settings, then turn the switch a multiple of 3 times. Again, this will preserve the state, and this is why multiples are in some sense special.
Finally, as for infinity: I'm with you in that it gets a bit uncomfortable to talk about the evenness or oddness of infinity itself. At that point it really comes down to the choice of definitions, and a perfectly reasonable definition is that infinity isn't a number but an unattainable goal (it's the trip, not the destination), in which case the concepts of evenness and oddness don't apply at all.