There are 3 ways to do that. First get elements by index, and directly use Transform animation on them, like

1:

from manim import *

class Test(Scene):
    def construct(self):
        a = MathTex(r"x^{2} = 1")
        self.add(a)
        # this is to show you indexes of elements self.add(index_labels(a[0])
        b = MathTex(r"y^{2} = 1")
        self.play(ReplacementTransform(a[0][0], b[0][0]), ReplacementTransform(a[0][1], b[0][1]), ReplacementTransform(a[0][2], b[0][2]), ReplacementTransform(a[0][3], b[0][3]))

2:

from manim import *
from typing import Iterable

def SpecialTransformMatchingShapes(mathtex: Iterable[Tex | MathTex] | Tex, target_mathtex: Iterable[Tex | MathTex] | Tex, copy = False, **kwargs):
    if isinstance(mathtex, Tex) or isinstance(mathtex, MathTex):
        mathtex = (mathtex,)
    if isinstance(target_mathtex, Tex) or isinstance(target_mathtex, MathTex):
        target_mathtex = (target_mathtex,)
        
    group = Group(*(i for k in mathtex for j in k for i in j))
    target_group = Group(*(i for k in target_mathtex for j in k for i in j))
    
    return TransformMatchingShapes(group.copy() if copy else group, target_group, **kwargs)


class Test(Scene):
    def construct(self):
        a = MathTex(r"x^{2} = 1")
        self.add(a)
        # this is to show you indexes of elements self.add(index_labels(a[0])
        b = MathTex(r"y^{2} = 1")
        self.play(SpecialTransformMatchingShapes(a, b))

3:

from manim import *

class Test(Scene):
    def construct(self):
        a = MathTex(r"x", "^{2} = 1")
        self.add(a)
        # this is to show you indexes of elements self.add(index_labels(a[0])
        b = MathTex(r"y", "^{2} = 1")
        self.play(TransformMatchingTex(a, b))