Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem.

The proof for abelian varieties is almost the same as that for elliptic curves. You just need to observe that if A,A' are two g-dimensional abelian varieties over F_q, then the following are equivalent:

(i) They are $F_q$-rationally isogenous.
(ii) They have the same characteristic polynomial of Frobenius.
(iii) For all $1 \leq i \leq g, \ |A(F_{q^i})| = |A'(F_{q^i})|$.
(iv) The Hasse-Weil zeta functions of A and A' coincide.

Although I have not checked in order to answer this question, I think it is likely that proofs -- or references to proofs -- of this fact can be found in at least one of the papers

Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521--560.

Waterhouse, W. C.; Milne, J. S. Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53--64. Amer. Math. Soc., Providence, R.I., 1971.

Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem [and, as Kevin Buzzard points out, of the semisimplicity of the Galois action on the Tate module, also proved by Faltings.]

The proof for abelian varieties is almost the same as that for elliptic curves. You just need to observe that if A, A' are two g-dimensional abelian varieties over F_q, then the following are equivalent:

(i) They are $F_q$-rationally isogenous.
(ii) They have the same characteristic polynomial of Frobenius.
(iii) For all $1 \leq i \leq g, \ |A(F_{q^i})| = |A'(F_{q^i})|$.
(iv) The Hasse-Weil zeta functions of A and A' coincide.

Although I have not checked in order to answer this question, I think it is likely that proofs -- or references to proofs -- of this fact can be found in at least one of the papers

Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521--560.

Waterhouse, W. C.; Milne, J. S. Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53--64. Amer. Math. Soc., Providence, R.I., 1971.

Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem.

The proof for abelian varieties is almost the same as that for elliptic curves. You just need to observe that if A,A' are two g-dimensional abelian varieties over F_q, then the following are equivalent:

(i) They are $\mathbb{F}_q$-rationally isogenous.
(ii) They have the same characteristic polynomial of Frobenius.
(iii) For all $1 \leq i \leq g, \ \#A(\mathbb{F}_{q^i}) = \#A'(\mathbb{F}_{q^i})$.
(iv) The Hasse-Weil zeta functions of A and A' coincide.

Although I have not checked in order to answer this question, I think it is likely that proofs -- or references to proofs -- of this fact can be found in at least one of the papers

Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521--560.

Waterhouse, W. C.; Milne, J. S. Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53--64. Amer. Math. Soc., Providence, R.I., 1971.

Yes, the local-global principle for isogenies is valid for all abelian varieties over all number fields, as a consequence of Faltings' isogeny theorem.

The proof for abelian varieties is almost the same as that for elliptic curves. You just need to observe that if A,A' are two g-dimensional abelian varieties over F_q, then the following are equivalent:

(i) They are $F_q$-rationally isogenous.
(ii) They have the same characteristic polynomial of Frobenius.
(iii) For all $1 \leq i \leq g, \ |A(F_{q^i})| = |A'(F_{q^i})|$.
(iv) The Hasse-Weil zeta functions of A and A' coincide.

Although I have not checked in order to answer this question, I think it is likely that proofs -- or references to proofs -- of this fact can be found in at least one of the papers

Waterhouse, William C. Abelian varieties over finite fields. Ann. Sci. École Norm. Sup. (4) 2 1969 521--560.

Waterhouse, W. C.; Milne, J. S. Abelian varieties over finite fields. 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969), pp. 53--64. Amer. Math. Soc., Providence, R.I., 1971.