Edit 3
I am not explaining the logic again, as it is explained below, but this is much cleaner.
This is the second example
list = {{x1, ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, 1 <= t <= 5]}, {y2,
ConditionalExpression[1, t >= 1 && t <= 5]}, {y1,
ConditionalExpression[1, t >= 1]}};
rule = {ConditionalExpression[1, 1 <= t <= 5] -> x,
ConditionalExpression[1, t >= 1 && t <= 5] -> y};
invrule = Map[Reverse, rule];
Select[list /. rule, ContainsAny[Join[{x, y}]]] /. invrule

Edit 2
In more complicated situations, where the rules do not combine nicely, you can create two separate ones, select them individually and join the list. Like so:
list = {{x1, ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, 1 <= t <= 5]}, {y2,
ConditionalExpression[1, t >= 1 && t <= 5]}, {y1,
ConditionalExpression[1, t >= 1]}};
rule = {ConditionalExpression[1, 1 <= t <= 5] -> x,
ConditionalExpression[1, t >= 1 && t <= 5] -> y};
invrule = Map[Reverse, rule];
then
Join[Select[list /. rule, #[[2]] == x &],
Select[list /. rule, #[[2]] == y &]] /. invrule

Edit 1
You can use the following to invert the rule
list = {{x1, ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, t >= 1]}};
rule = {ConditionalExpression[1, 1 <= t <= 5] -> x};
invrule = Map[Reverse, rule];
and then
Select[list /. rule, #[[2]] == x &] /. invrule

And for the second example
list = {{x1, ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, 1 <= t <= 5]}, {y2,
ConditionalExpression[1, t >= 1 && t <= 5]}, {y1,
ConditionalExpression[1, t >= 1]}};
rule = {ConditionalExpression[1, 1 <= t <= 5] -> x,
ConditionalExpression[1, t >= 1 && t <= 5] -> x};
invrule = Map[Reverse, rule];
and then
Select[list /. rule, #[[2]] == x &] /. invrule
gives

Original
@Nasser gave one way, here's another. Set the condition you want to pick to be x
using a rule.
list = {{x1, ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, 1 <= t <= 5]}, {y1,
ConditionalExpression[1, t >= 1]}};
rule = ConditionalExpression[1, 1 <= t <= 5] -> x;
invrule = x -> ConditionalExpression[1, 1 <= t <= 5];
and then
Select[list /. rule, #[[2]] == x &] /. invrule
