Its white coat hypertension, where you have a higher BP when recorded at a clinic or hospital setting, which is due to anxiety as you mentioned, but is within normal limits at comfortable settings.
As you said the readings are near 130/80 mmhg, and hoping you are taking your hypertension medications regularly, it's normal.
As your doctor also mentioned these spikes are normal, there is nothing to worry about.
For your satisfaction, you can start recording your BP atleast two times a day, at home, in a comfortable setting, after a proper rest, sitting relaxed for atleast 15 to 20 mins, not to be done immediately after doing some exhausting activities, or after a lot of walking.
Record the BP with a correct method, to avoid unreliable readings.
If your BP reading are any time more than 160 above or even more than 100 below, recheck BP after 10 to 15 mins again after proper rest, till it returns within normal limits.
Anytime the reading in more than 180 above or even more than 120 below, Make sure you recheck again after 1o to 15 mins, still if similar, visit a nearby clinic or hospital immediately.
Irrespective of the BP readings, if any time there are complaints of severe headcahe, sudden neck pain and stiffness, blurring of vision, fainting episode or dizziness, difficulting balancing, difficulty using or sudden weakness/ tingling sensation in arms, legs, face, nausea vomiting, visit a nearby clinic/ hospital nearby.
Note down the recordings for the day for 1 week, if any abnormal values noted continously, despite taking medications normally, visit your doctor with them.
Next Steps
Consult with a general physician or practioner online or in a clinic.
You can connect with me on practo or via call/ WhatsApp (915093049 eight), for furthur queries.