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John is right. I'll paraphrase his answer for the benefit of future users.

Let's dissect this :

var('u_5 u beta')

This is equivalent to u_5, u, beta)=SR.var("u_5 u beta). In particular, this

create a symbolic variable $\beta$ (a Python object defined by Sage), a symbolic variable $u_5$ and a *symbolic variable $u$, then
sets Python variables :

+beta pointing to $beta$,
- u_5 pointing to $u_5$, and
- u pointing to $u$.
y = u - u_5 + beta*(exp(u)-u-4)

This create a *symbolic expression, a Python structure defuined by sage, containing, among others, a pointer to $beta$.

beta = 1.0
u_5 = -4.0

This binds beta and u_5 to the numerical values 1.0 and -4 respectively. The links to $beta$ and $u_5$ are lost. Thr expression bound to p is *unchanged* : the pointers to $beta$ and $u_5$ still point to the respective symbolic variables.

p = y.subs(u==u_5)

This create a new symbolic expression where the pointer to $u$ us replaced by the value of the Python variable u_5, which is now the numeric value -4.0. The pointers to $beta$ and $u_5$ are unmodified.

This prints the new symbolic expression, which still contains (pointers to) symbolic variables.what you wnt to do is probably :

sage: y.subs(u==u_5).subs([beta==1, u_5==-4])
e^(-4)

or, more probably :

sage: y.subs(u==u_5).subs([beta==1, u_5==-4]).n()
0.0183156388887342

This can be abbreviated as :

sage: y.subs(u==u_5)(beta=1, u_5=-4).n()
0.0183156388887342

See the documentation for details, and the initial chapters of this marvelous book, which I cannot recommend too much.

Note that :

sage: y.subs([u==u_5, beta==1, u_5==-4])
e^u_5

in which the substitutions are executed in a different order, resulting in a different value.

HTH,